^ -r^io u Digitized by the Internet Archive in 2010 with funding from Boston Library Consortium Member Libraries http://www.archive.org/details/elementarycourseOOtann Al^ ELEMENTARY COURSE IN ANALYTIC GEOMETRY BY Q J. H. TANNER ASSISTANT PROFESSOR OF MATHEMATICS IN CORNELL UNIVERSITY AND JOSEPH ALLEN FORMERLY INSTRUCTOR IN MATHEMATICS IN CORNELL UNIVERSITY TUTOR IN THE COLLEGE OF THE CITY OF NEW YORK BOSTON COLLEGE LIFKAHT ^:.o_ CHESTOTT HILL, M^.^. MATH, DEPT, NEW YORK •:• CINCINNATI •;• CHICAGO AMERICAN BOOK COMPANY Copyright, 1898, by J. H. TANNER axd JOSEPH ALLEN. ANA. GEOM. W. P. I 15055ii THE CORNELL MATHEMATICAL SERIES LUCIEN AUGUSTUS WAIT • • • General Editor (SENIOE PROFESSOR OF MATHEMATICS IN CORNELL UNIVERSITY) The Cornell Mathematical Series, lucien augustus wait, (Senior Professor of Mathematics in Cornell University,) GENERAL EDITOR. This series is designed primarily to meet tlie needs of students in En- gineering and Architecture in Cornell University ; and accordingly many practical problems in illustration of the fundamental principles play an early and important part in each book. While it has been the aim to present each subject in a simple manner, yet rigor of treatment has been regarded as more important than simplicity, and thus it is hoped that the series will be acceptable also to general students of Mathematics. The general plan and many of the details of each book were discussed at meetings of the mathematical staff. A mimeographed edition of each vol- ume was used for a term as the text-book in all classes, and the suggestions thus brought out were fully considered before the work was sent to press. The series includes the following works : ANALYTIC GEOMETRY. By J. H. Tanner and Joseph Allen. DIFFERENTIAL CALCULUS. By James 'McMahon and Virgil Snyder. INTEGRAL CALCULUS. By D. A. Murray. PREFACE Although in the writing of this book the needs of the students in the various departments of Engineering and of Architecture in Cornell University have received the first consideration, care has also been taken to make the work suitable for the general student and at the same time useful as an introduction to a more advanced course for those students who may wish to specialize later in mathematics. Among the features of the book are : (1) An extended introduction (Chaps. II, III, IV), in which it is hoped that the fundamental problems of the sub- ject are clearly set forth and sufficiently illustrated. The chief difficulty which the beginner in Analytic Geometry usually has to overcome is the relation between an equation and its locus ; having really mastered this, he easily and rapidly acquires a knowledge of the properties to which this relation leads, and especial care has therefore been given to this matter. Analytic Geometry is broader than Conic Sec- tions, and it is the firm conviction of the authors that it is far more important to the student that he should acquire a familiarity with the spirit of the method of the subject than that he should be required to memorize the various properties of any particular curve. (2) The making use of some intrinsic properties of curves (see Arts. 106, 112, 118), which experience with many classes has shown to give the student an unusually strong grasp on the equation of the second degree from Avhich the a:z/-term is absent. (3) Introduction of the demonstrations of general theorems by numerical examples. This not only makes clear to the student what is to be done, but shows also the method to be employed, — it generalizes after the student is acquainted with the particular. (4) Easy but rigorous proofs of all the theorems within the scope of the book. E.g., in Art. 67 it is proved, and vi PBEFACE very simply, too, that tlie vanishing of the discriminant is not only a necessary^ but also the sufficient condition that the quadratic equation represents a pair of straight lines. It may also be mentioned here that, in the early part of the book, two or more figures are given in connection with a proof and so lettered that the same demonstration applies to each. It is hoped that this will help to convince the student of the generality of the demonstration. A copious index which enables one almost instantly to turn to anything contained in the book has also been added. The engineering students at Cornell University study Analytic Geometry during the first term of their freshman year, and experience has shown that it is best to devote a few lessons at the beginning of the term to a rapid review of those parts of the Algebra and Trigonometry that are essential to the reading of the Analytic Geometry. The first twenty-three pages are devoted to this matter, and may, of course, be omitted by those classes that take up the subject immediately after reading the Algebra and Trigonometry. The book contains little more than can be mastered by a properly prepared student of average ability in from twelve to fourteen weeks ; if less than that time can be devoted to the work, the individual teacher will know best what parts may be most wisely omitted by his pupils. A list of lessons for a short course of eleven weeks is, however, suggested on the next two pages. A few specific acknowledgments of indebtedness are made in foot-notes in the appropriate places in the book. Of the large number of examples which are inserted, many are origi- nal, while many others have come to be so common in text- books that no specific acknowledgment for them can be made. We take great pleasure in expressing here our thanks to the other authors of this series of books for their many helpful suggestions and criticisms ; to our colleagues. Dr. J. I. Hutchinson and Dr. G. A. Miller, who have so greatly assisted us in reading the proof, and the latter of whom also read the manuscript before it went to press ; to Mr. Peter Field, Fellow in Mathematics, and Mr. E. A. Miller for solving the entire list of examples ; and to Mr. V. T. Wilson, Instructor in Drawing in Sibley College, for the care with which he has made the figures. LIST OF LESSONS SUGGESTED FOR A SHORT COURSE [From the various sets of exercises the teacher is expected to make selec- tions lor each lesson. The fifth day of each week should be devoted to reviewing the preceding four lessons.] Lesson Pages Articles 1 . o . . . 1-9 ..... 1-8 2 . . . •. . 9-15 ..... 9-12 3 .... . 15-23 13-17 4 24-28 18-22 5 29-33 23-27 6 34-40 28-30 7 40-42 ..... 31 8 .... . 43-52 32-37 As far as " Exercises," p. 52. 9 52-57 ..... 38-41 10 .... . 58-60 11 61-65 42-45 12 65-73 46-48 With examples selected from p. 79. 13 .... . 73-80 ..... 49 14 81-85 50-53 15 86-94 54-58 16 94-98 ..... 59-61 17 98-104 62-63 18 .... . 105-110 ..... 64-65 19 .... I 11^-11^ . ... 66, 67, 69 1 118-119 ' ' 20 ... . .119-122 2^ r 123-127 r 70-72 1 129-131 • • • • 1 75-76 vii viii LESSONS FOB SHORT COURSE Lesson Pages Articles 22 131-137 77-78 23 137-142 79-82 24 142-149 83-85 25 149-155 ..... 86-90 2Q 156-165 93-100 27 165-169 28 .... . 170-177 101-107 29 179-186 109-112 ^« • • • • [f^Z- ■ ■ '''''''-''' 31 195-202 118-122 32 203-208 123-126 33 209-216 127-132 34 216-218 35 219-225 133-137 36 225-233 138-140 37 235-242 142-145 33 f 242-247 ( 146-148 ' ' * ' 1 250-254 • • • • j 152-154 39 .... . 254-264 155-157 40 265-272 160-164 41 272-283 165-170 42 284-291 171-174 43 292-298 175-177 44 309-330 185-198 CONTENTS PART I.— PLANE ANALYTIC GEOMETRY CHAPTER I Introduction Algebraic and Trigonometric Conceptions AKTICLB PAGE 1. Number 1 2. Constants and variables 2 3. Functions 3 4. Identity, equation, and root 4 5. Functions classified 4 6. Notation 5 7. Continuous and discontinuous functions 6 8 ) ' y The quadratic equation. Its solution 9 10. Zero and infinite roots ........ 11 11. Properties of the quadratic equation 12 12. The quadratic equation involving two unknowns . . .13 Trigonometric Conceptions and Formulas 13. Directed lines. Angles 15 14. Trigonometric ratios ......... 17 15. Functions of related angles 18 16. Other important formulas 19 17. Orthogonal projection . .21 CHAPTER II Geometric Conceptions. The Point I. Coordinate Systems 18. Coordinates of a point 24 19. Analytic Geometry 25 ix CONTENTS AETICLE 20. Positive and negative coordinates 21. Cartesian coordinates of points in a plane . 22. Rectangular coordinates . . . . 23. Polar coordinates , . . . . 24. Notation 25 26 :[ 27. 28. 29. 30. 31. II. Elementary Applications Distance between two points (1) Polar coordinates (2) Cartesian coordinates ; axes not rectangular . (3) Rectangular coordinates .... Slope of a line Summary . The area of a triangle (1) Rectangular coordinates .... (2) Polar coordinates To find the coordinates of the point which divides, in a ratio, the straight line from one given point to another Fundamental problems of analytic geometry given PAGE 25 26 27 29 30 31 32 33 33 34 34 36 37 40 CHAPTER III The Locus of an Equation 32. The locus of an equation .... 33. Illustrative examples : Cartesian coordinates 34. Loci by polar coordinates .... 35. The locus of an equation .... 36. Classification of loci 37. Construction of loci. Discussion of equations 38. The locus of an equation remains unchanged : (a) by any trans- position of the terms of the equation ; and (^) by multiply- ing both members of the equation by any finite constant 39. Points of intersection of two loci 40. Product of two or more equations 41. Locus represented by the sum of two equations CHAPTER IV The Equation of a Locus 42. The equation of a locus ........ 43. Equation of straight line through two given points . 43 43 46 47 48 49 52 53 54 56 61 61 CONTENTS XI ARTICLE PAGE 44. Equation of straight line through given point and in given 63 64 65 66 67 73 direction 45. Equation of a circle ; polar coordinates 46. Equation of locus traced by a moving point 47. Equation of a circle : second method 48. The conic sections ..... 49. The use of curves in applied mathematics . CHAPTER V The Straight Line. Equation of First Degree Ax + By+C = 50. Recapitulation 81 51. Equation of straight line through two given points . . .81 52. Equation of straight line in terms of the intercepts which it makes on the coordinate axes ....... 83 53. Equation of straight line through a given point and in a given direction 84 54. Equation of straight line in terms of the perpendicular from the origin upon it, and the angle which that perpendicular makes with the a:-axis . . . . . . . .86 55. I^ormal form of equation of straight line : second method . 87 56. Summary ........... 88 57. Every equation of the first degree between two variables has for its locus a straight line 89 58. Reduction of the general equation Ax -{ By -\- C = to the standard forms. Determination of a, b, m, jj, and a in terms of .4, A and C 91 59. To trace the locus of an equation of the first degree . . .94 60. Special cases of the equation of the straight line Ax-]-By+ C = 95 61. To find the angle, made by one straight line with another . . 97 62. Condition that two lines are parallel or perpendicular . . 98 63. Line which makes a given angle with a given line . . .101 64. The distance of a given point from a given line . . . 105 65. Bisectors of the angles between two given lines . . . 108 66. The equation of two lines 110 67. Condition that the general quadratic expression may be factored 111 68. Equations of straight lines : coordinate axes oblique . . .115 69. Equations of straight lines : polar coordinates .... 118 xii CONTENTS CHAPTER VI Transformation of Coordinates ARTICLE PAGE 70. Introductory 123 I. Cartesian Coordinates Only 71. Change of origin, new axes parallel respectively to the original axes 124 72. Transformation from one system of rectangular axes to another system, also rectangular, and having the same ori- gin ; change of direction of axes 126 73. Transformation from rectangular to oblique axes, origin un- changed ........... 127 74. Transformation from one set of oblique axes to another, origin unchanged . . . . . . . . . . 128 75. The degree of an equation in Cartesian coordinates is not changed by transformation to other axes . . . . 129 ^ II. Polar Coordinates 76. Transformations between polar and rectangular systems . . 130 CHAPTER VII The Circle Special Equation of the Second Degree Ax"^ + Aif + 2 Gx -V 2 Fy ^ C = 77. Introductory 135 78. The circle : its definition and equation 135 79. In rectangular coordinates every equation of the form x^ 4- y^ + 2Gx-\-2Fy-]-C = represents a circle .... 137 80. Equation of a circle through three given points . . . 138 Secants, Tangents, and Normals 81. Definitions of secants, tangents, and normals . . . . 140 82. Tangents : IllustratiA^e examples ...... 141 83. Equation of tangent to the circle x^ + y'^ = r^ in terms of its slope 142 84. Equation of tangent to the circle in terms of the coordinates of the point of contact : the secant method .... 144 CONTENTS Xlll 85. 86. 87. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. PAGE Equation of a normal to a given circle . . , . . 147 Lengths of tangents and normals. Subtangents and sub- normals ........... 149 Tangent and normal lengths, subtangent and subnormal, for the circle 150 To find the length of a tangent from a given external point to a given circle . . . . . . . • .151 From any point outside of a circle two tangents to the circle can be drawn . 152 Chord of contact ......... 154 Poles and polars 156 Equation of the polar . . . . . . . . 156 Fundamental theorem 157 Geometrical construction for the polar of a given point, and for the pole of a given line, with regard to a given circle . 158 Circles through the intersections of two given circles . . 160 Common chord of two circles ....... 160 Radical axis ; radical center ....... 161 The equation of a circle : polar coordinates .... 162 Equation of a circle referred to oblique axes .... 163 The angle formed by two intersecting curves .... 164 CHAPTER VIII The Conic Sections 101. Recapitulation 170 I. The Parabola Special Equation of Second Degree Ax^ + 2Gx + 2Fi/ + C = 0, ov Bf + 2Gx-h2Fu -^ a = 102. The parabola defined 103. First standard form of the equation of the parabola 104. To trace the parabola y'^ = 4:px .... 105. Latus rectum ........ 106. Geometric property of the parabola. Second standard equa- tion ........... 107. Every equation of the form Ax^-\-2Gx + 2Fi/ + C = 0, or By^ + 2 Gx ^ 2 Fy + C = 0, represents a parabola whose axis is parallel to one of the coordinate axes ......... 108. Reduction of the equation of a parabola to a standard form . 170 171 172 173 173 175 177 xiv CONTENTS II. The Ellipse Special Equation of the Second Degree Ax^ + Bif-^2Gx + 2Fi/+C = ARTICLE PAGE 109. The ellipse defined . .179 110. The first standard equation of the ellipse .... 180 111. To trace the ellipse ^4-^=1 182 112. Intrinsic property of the ellipse. Second standard equation . 183 113. Every equation of the form Ax^ + Bi/''-}-2Gx-\-2Fi/ ■}- C = represents an ellipse whose axes are parallel to the coordi- nate axes, if A and B have the same sign .... 186 114. Reduction of the equation of an ellipse to a standard form . 189 III. The Hyperbola Special Equation of the Second Degree Ax'^ - Bi/-^-]-2Gx + 2Fy + C = 115. The hyperbola defined 190 116. The first standard form of the equation of the hyperbola . 191 117. To trace the hyperbola —- ^ = 1 193 118. Intrinsic property of the hyperbola. Second standard equa- tion 195 119. Every equation of the form Ax'^ -\-By^ + 2Gx + 2Fy ■]- C = represents an hyperbola whose axes are parallel to the coor- dinate axes, if A and B have unlike signs .... 197 120. Summary 199 TV. Tangents, Normals, Polars, Diameters, etc. 121. Introductory 200 122. Tangent to the conic Ax"^ + By'^ + 2 Gx ^2 Fy + C = in terms of the coordinates of the point of contact : the secant method 200 123. Normal to the conic Ax'^ + By^ + 2 Gx + 2 Fy + C = 0, at a given point . 203 124. Equation of a tangent, and of a normal, that pass through a given point which is not on the conic ..... 205 CONTENTS XV 125. Through a given external point two tangents to a conic can be drawn 206 ^26. Equation of a chord of contact 207 127. Poles and polars 209 128. Fundamental theorem 210 129. Diameter of a conic section 211 130. Equation of a conic that passes through the intersections of two given conies 213 V. Polar Equation of the Conic Sections 131. Polar equation of the conic ..... 132. From the polar equation of a conic to trace the curve 214 215 CHAPTER IX The Parabola y'^ — ^px 133. Review 219 134. Construction of the parabola 220 135. The equation of the tangent to the parabola y'^ ^'^px in terms of its slope 221 136. The equation of the normal to the parabola y^ = ^px in terms of its slope 222 137. Subtangent and subnormal. Construction of tangent and normal 222 . 138. Some properties of the parabola which involve tangents and normals . . . . . . . . . - . 225 139. Diameters 230 140. Some properties of the parabola involving diameters . . 232 141. The equation of a parabola referred to any diameter and the tangent at its extremity as axes .... 233 CHAPTER X The Ellipse ^ + ^ = 1 a2 62 142. Review 237 143. The equation of the tangent to the ellipse [- ^ = 1 in a^ }p- terms of its slope 238 144. The sum of the focal distances of any point on an ellipse is constant; it is equal to the major axis . . . 239 xvi CONTENTS ARTICLE PAGE 145. Construction of the ellipse 240 146. Auxiliary circles. Eccentric angle 242 147. The subtangent and subnormal. Construction of tangent and normal .......... 244 148. The tangent and normal bisect externally and internally, respectively, the angles between the focal radii of the point of contact .......... 246 149. The intersection of the tangents at the extremity of a focal chord 247 150. The locus of the foot of the perpendicular from a focus upon a tangent to an ellipse ........ 248 151. The locus of the intersection of two perpendicular tangents to the ellipse . 249 152. Diameters 250 153. Conjugate diameters 252 154. Given an extremity of a diameter, to find the extremity of its conjugate diameter 253 155. Properties of conjugate diameters of the ellipse . . . 254 156. Equi-con jugate diameters 257 157. Supplemental chords 259 158. Equation of the ellipse referred to a pair of conjugate diam- eters 260 159. Ellipse referred to conjugate diameters ; second method . 261 CHAPTER XI The Hyperbola - — '^ = 1 160. Review 265 161. The difference between the focal distances of any point on an hyperbola is constant ; it is equal to the transverse axis . 266 162. Construction of the hyperbola 267 163. The tangent and normal bisect internally and externally the angles between the focal radii of the point of contact . . 268 164. Conjugate hyperbolas 270 165. Asymptotes . . . . . - 272 166. Relation between conjugate hyperbolas and their asymptotes 275 167. Equilateral or rectangular hyperbola 277 168. The hyperbola referred to its asymptotes .... 278 169. The tangent to the hyperbola xy = c^ 280 170. Geometric properties of the hyperbola ..... 281 CONTENTS XVll 171. Diameters 284 172. Properties of conjugate diameters of the hyperbola . . 285 173. Supplemental chords 287 174. Equations representing an hyperbola, but involving only one variable 288 CHAPTER XII General Equation of the Second Degree Ax^ + 2 Hxy + Bf + 2Gx-h2Fi/+ C=zO 175. General equation of the second degree in two variables . . 292 176. Illustrative examples 294 177. Test for the species of a conic 297 178. Center of a conic section 298 179. Transformation of the equation of a conic to parallel axes through its center , 299 180. The invariants A + B andH^ - AB 301 181. To reduce to its simplest standard form the general equation of a conic 303 182. Summary 306 183. The equation of a conic through given points . . . 307 184. Definitions CHAPTER XIII Higher Plane Curves 309 I. Algebraic Curves 185. The cissoid of Diodes 186. The conchoid of Nicomedes 187. The witch of Agnesi 188. The lemniscate of Bernouilli 189 a. The lima9on of Pascal 189 6. The cardioid . 190. The Neilian, or semi-cubical parabola II. Transcendental Curves 191. The cycloid 192. The hypocycloid 309 312 314 315 318 319 320 321 323 XVlll CONTENTS III. Sjnrals AETICLE 193. Definition 194. The spiral of Archimedes 195. The reciprocal, or hyperbolic, spiral 196. The parabolic spiral . . . . 197. The lituus or trumpet 198. The logarithmic spiral 325 325 326 328 328 329 PART II. — SOLID ANALYTIC GEOMETRY CHAPTER I 199. 200. 201. 202. 203. 204. 205. 206. 207. Coordinate Systems. The Point Introductory 331 . 332 . 333 . 333 . 334 Rectangular coordinates ........ Polar coordinates ......... Relation between the rectangular and polar systems Direction angles : direction cosines . . . ... Distance and direction from one point to another; rectangu- lar coordinates ......... The point which divides in a given ratio the straight line from one point to another Angle between two radii vectores. Angle between two lines Transformation of coordinates ; rectangular systems 336 337 338 339 CHAPTER II The Locus of an Equation. Surfaces 208. Introductory 342 209. Equations in one variable. Planes parallel to coordinate planes 343 210. Equations in two variables. Cylinders perpendicular to coor- dinate planes 344 211. Equations in three variables. Surfaces 346 212. Curves. Traces of surfaces 347 213. Surfaces of revolution . . . - 348 CONTENTS XIX CHAPTER III Equations of the First Degree Ax -{- Bij -{■ Cz + D = 0. Planes AND Straight Lines I. The Plane ARTICLE PAGE 214. Every equation of the first degree represents a plane . . 353 215. Equation of a plane through three given points . . . 354 216. The intercept equation of a plane 354 217. The normal equation of a plane ...... 355 218. Reduction of the general equation of first degree to a stand- ard form. Determination of the constants a, 6, c, p, a, jS, y 356 219. The angle between two planes. Parallel and perpendicular planes 357 220. Distance of a point from a plane 359 II. The Straight Line 221. Two equations of the first degree represent a straight line . 359 222. Standard forms for the equations of a straight line ^ (a) The straight line through a given point in a given direction 360 (b) The straight line through two given points . . . 360 (c) The straight line with given traces on the coordinate planes 361 223. Reduction of the general equations of a straight line to a standard form. Determination of the direction angles and traces I. Third standard form : traces ...... 362 II. First standard form : direction angles .... 362 224. The angle between two lines ; between a plane and a line . 363 CHAPTER IV Equations of the Second Degree. Quadric Surfaces 225. The locus of an equation of second degi'ee .... 367 226. Species of quadrics. Simplified equation of second degree . 368 227. Standard forms of the equation of a quadric .... 370 2 2 2 228. The ellipsoid : equation -^ + ^ + % = l . . . .371 a^ 0^ c^ XX COIiTENTS ARTICLE 229. The un-parted hyperboloid : equation -2 + ^ 230. The bi-parted hyperboloid: equation -^~jo — 231. The paraboloids : equation ^ ± ^ = 2 ~2 X y 232. The cone: equation — „ + t; o/' -. = 233. The hyperboloid and its asymptotic cone 373 375 376 378 379 APPENDIX Note A. Note B. Note C. Note D. Note E. Note F. Answers Index Historical sketch Construction of any conic .... SjDecial cases of the conies .... Every section of a cone by a plane is a conic . Parabola as a limiting form of ellipse or hyperbola Confocal conies 381 382 383 384 387 388 391 000 AI^ALYTIC GEOMETRY PART I CHAPTER I INTRODUCTION ALGEBRAIC AND TRIGONOMETRIC CONCEPTIONS 1. Number. A number is most simply interpreted as expressing the measurement of one quantity by another quantity of the same kind first chosen as a unit of measure ; it is positive, or +, if the measuring unit is taken in the same sense as the thing measured; and negative, or — , if this measuring unit is taken in the opposite sense. I^.g.^ the unit dollar may be regarded as a dollar of assets, or as a dollar of liabilities ; if it is regarded as a dollar of assets, then assets measured by it produce positive numbers, while liabilities measured by it produce negative numbers. The above definition is consistent with the one usually given; viz. that numbers are positive or negative according as they are greater or less than zero. If tlie operations of addition, subtraction, multiplication, division, raising to integer powers, extracting roots, or any combination of these operations, are performed upon given numbers, the result in every case is a number ; it is imaginary TAN. AX. GEOJI. — 1 2 ANALYTIC GEOMETRY [Ch. I. if it involves in any way whatever an indicated even root of a negative number; otherwise it is real. Every imaginary number may be reduced to the form a + 6 V— 1, where a and h are real, and b =^0. 2. Constants and variables. If AB and AC are two given straight lines making an angle a at the point A, and if any tAvo points X and F, on these lines, respectively, ^ ^ are joined by a straight line, then Area of triangle AXY =\' AX- A Y • sin a, i.e.^ A = ^ ' x- ^ ' sin a^ where x is the length of AX, y is the length of A Z, and A is the area of the triangle. If now the points X and Y are moved along the lines AB and AC vn any way whatever, then A, x, and y will each pass through a series of different values, — they are variable num- bers or variables ; while \ and sin « will remain unchanged, — they are constant numbers or constants. It is to be remarked that \ has the same value wherever it occurs, — it is an absolute constant; while «, though constant for this series of triangles, may have a different constant value for another series of triangles, — it is an arbitrary constant. Because x and y may separately take any values what- ever they are independent variables; while A, whose value depends upon the values of x and ?/, is a dependent variable. The illustrations just given may serve to give a clearer conception of the following more formal definitions. An absolute constant is a number which has the same value 2 wherever it occurs; such are the numbers 2, 7, -|, 6% tt, e 1-3. ] INTR OD UCTION 3 (where ir = 3.14159265..-, approximately -y-, the ratio of the circumference of a circle to its diameter; and ^ = 2.71828182... = 1 + ^ + |-+ -|+ ..., approximately -y-, the base of the Naperian system of loga- rithms). An arbitrary constant is a number which retains the same value throughout the investigation of a given problem, but may have a different fixed value in another problem. An independent variable is a number that may take any value whatever within limits prescribed by the conditions of the problem under consideration. A dependent variable is a number that depends for its value upon the values assumed by one or more independent variables.* A number that is greater than any assignable number, however great, is an infinite number; one that varies and becomes and remains smaller (numerically, not merely alge- braically less) than any assigned number, however small, is an infinitesimal number. All other numbers are finite. 3. Functions. A number so related to one or more other numbers that it depends upon these for its value, and takes in general a definite value, or a finite number of definite values, when each of these other numbers takes a definite value, is a function of these other numbers. -E^.^., the cir- cumference and the area of a circle are functions of its radius; the distance traveled by a railway train is a function of its time and rate; if ?/ = 3 a;^ -f- 5iK — 8, then ?/ is a function of x. * All these kinds of numbers will be met and better illustrated in succeed- ing chapters of this book. E.g.^ see Art. 65, Note. 4 ANALYTIC GEOMETRY [Ch. 1. 4. Identity, equation, and root. If two functions involv- ing the same variables are equal to each other for all values of those variables they are identically equal. Such an equality is expressed by writing the sign = between the two functions, and the exj)ression so formed is an identity. If, on the other hand, the two functions are equal to each other only for particular values of the variables, the equality is expressed by writing the sign = between the two func- tions, and the expression so formed is an equation. The particular values for which the two functions are equal, i.e., those values of the variables which satisfy the equation, are the roots of the equation. E.g., {x + yy^=x^- -\-2xy + y% (x + a)(x - a) + a^ = x% , , 3 .r^ — a: + 3 and X H = x—1 x—1 are identities ; while 3 x^ — 10 x + 2 = 2x^ - 4:X — 6, or, what is the same thing, a;2 - 6 X + 8 = 0, is an equation. The roots of this equation are the numbers 2 and 4. Special attention is called to the fact that an equation always imposes a condition. E.g., x2 — 6x + 8 = if, and only if , x = 2 or x = 4. So also the equa- tion ax -\- hy -{■ c = ^ imposes the condition that x shall be equal to -by-c ^ a 5. Functions classified. A functional relation is usually expressed by means of an equation involving the related numbers. If the form of this equation is such that one of the variables is expressed directly in terms of the others, then that variable is called an explicit function of the others; if it is not so expressed, it is an implicit function. E.g., the equations y = V o — x% x^ -}- y'^ = 5, and x = V5 — y'^ express the same relation between x and y ; in the first y is an explicit function 4-6.] INTttODtWnon 5 of X, in the second each is an implicit function of the other, while in the third :*: is an explicit function of y. The word "function" is, for brevity, usually represented by a single letter, such as /, jF, (/>, i|r,... ; thus y = <^(x) means that ^ is a function of the independent variable x, and is read "^ equals the (/)-f unction of a:"; so also z = F(ii, v, x) means that 2 is a function of the independent variables u, v, and X, and it is read, "2 equals the ^-function of w, v, and x.'' A function is algebraic if it involves, so far as the inde- pendent variables are concerned, only a finite number of the operations of addition, subtraction, multiplication, division, raising to integer powers, and extracting roots. All other functions are transcendental. E.q., 2 x^ — 5 a: — 17, rrv + v^ — 7x, and ^-, are algebraic if i •> i) i) X ^ xy — 1 y- ° functions; while 2^, (f, sin a;, tan~^2, and log < are transcendental func- tions. 6. Notation. In general, absolute constants are repre- sented by the Arabic numerals, while arbitrary constants and variables are represented by letters. A few absolute con- stants are, however, by general consent, represented by let- ters ; examples of such constants are tt and e (Art. 2). Variables are usually represented by the last letters of the alphabet, such as w, v, w^ x^ y^ z \ while the first letters, a^h^c^"' are reserved to represent constants. Particular fixed values from among those that a variable may assume are sometimes in question; e.^., the values, a: = 2 and 2; = — 1, for which the function a;^— 2: — 2 vanishes; such values may conveniently be denoted b}^ affixing a sub- script to the letter representing the variable. Thus x^^ x^, ^31 * * * will be used to denote particular values of the variable x. Similarly, variables which enter a problem in analogous 6 ANALYTIC GEOMETRY [Ch. I. ways are usually denoted by a single letter having accents attached to it ; thus a;', x'^ ^ x'", ••• denote variables that are similarly involved in a given problem. Again, each of the two equations, ^=Bx^ — 4:x-]-10 and y = <^(a;), asserts that y is a function of x ; but while the former tells precisely how y depends upon x^ the latter merely asserts that there is such a dependence, without giving any information concerning the form of that depend- ence. If several different forms of functions present them- selves in the same problem, they are represented by different letters, each letter representing a particular form for that problem, though it may be chosen to represent an entirely different form in another problem. E.g., if the form of cfi, in a given problem, is defined by the equation 3 x^ - x^+ 6 (x) = then, in the same problem, 2a,- + 1 ' <^(^0- ^t~r^^ - <^a) = L and cf>(0) = 5. 2 y + 1 o 7. Continuous and discontinuous functions. In general a function takes different values when different values are assigned to its independent variable. If ?/ = (^)? then, for X = a and x = d, the function becomes y-^ = (<^) and ^2 = <^(^), and ?/j is in general different from y^- The func- tion (j>(^x') is said to be a continuous function of x between x = a and x = b, if, while x is made to pass successively through all real values from a to 6, y remains real and finite and passes corresj)ondingly through all values from y^ to y^' This definition may be more precisely stated, thus : If a:^ and ajg are any real values of x which lie between the values a and b, and if the cor- responding values of y, viz. (xj) and (x.^, are real and finite; and if 6-7.] INTRODUCTION i a positive number rj can be found, such that by tailing, numerically, ^l — -^2 *^ 'Z' it will follow that, numerically, (x^) - (x2) < e, where e is any assigned positive number, however small ; then (x) is a continuous function of x for values from a to b. Or, in words : y h a continuous function of x for all values of x in the interval from a to b, if, by taking any two values of x in the interval sufficiently near together, the difference between the corresponding values of y can be made less than any assigned number, however small. A discontinuous function is one that does not fulfil the conditions for continuity. It is, however, usually/ discon- tinuous for only a limited number of particular values of its independent variable, while between these values it is con- tinuous. As familiar examples of continuous functions may be mentioned : the length of a solar shadow ; the area of a cross-section of a growing tree, or of a growing peach ; the height of the mercury in a barometer ; the temperature of a room at varying distances from the source of heat ; and interest as a function of time. So, also, y=3a;2-f4a; + l is a continuous function of x for all finite values of x. For, 7/ remains real and finite so long as x remains real and finite, and, if x-^ and X2 be any two finite values of x which differ from each other by 7;, i.e., if X2 = x^± 7/, then 1^2 - ?/i = 3 2:22 + 4 3^2 + 1 - (^ ^1^ + ^ ^1 + 1)' = 3(2:1 ± vY 4- 4(2^1 ± 7;) + 1 - (3:ri2 + 42:1 + 1), = ±(62:1-^4 + 377)7;. Now to show that ?/ = 3 2:^ + 4 2; + 1 is continuous for X = 2^1, it only remains to show that, by taking rj sufficiently 8 ANALYTIC GEOMETRY [Ch. I. small, ^.e., by taking x^ sufficiently near a^j, 7/^ can be made to differ from ?/^ by less than any assigned number (e), how- ever small. But this is evident; for y may be taken as near zero as desired, hence the factor 6 a;^ + 4 + 3 ?; as near 6 2:^ + 4 as desired, and the product therefore as near zero as is neces- sary to be less than e. On the other hand, if, at regular intervals of time, apples are dropped into a basket, the combined weight of the basket and apples will increase discontinuously ; ^.e., their total weight is a discontinuous function of the time. EXERCISES 1. li Ax -]- By -\-C = 0, prove that ?/ is a continuous function of x\ and X, of y. 2. If ,r2+ ?/2_ 4 _ 0^ prove that ?/ is a continuous function of x, when 2>x>-2. 3. If ^ -I- ^_ = 1, prove that ar is a continuous function of y, when b>y>-b. 4. If ^ — 1 = 0. is a; a continuous function of ?/ ? 5. If 61/ — 9 = 0, is 5 a continuous function of tl 6. If ifi — 3 y = 0, is i< a continuous function of v ? Is v a continu- ous function of u ? 7. Show that all functions of the form OoX" + a^.r"-^ + a2-2^«-2 + ••• + an-\X + ff„, where CTq, a^ a<)--' an are constants, are continuous for all finite values of X. 8. If " = 5^"\ show that y is discontinuous for x = 1. y ~ - 1 9. Find the value of x for which y, — c ^'"'^ ~ , is discontinuous. 7-9.] INTRODUCTION 9 10. Interest on money loaned is calculated by the formula I^ P-R'T. Is the interest (/) a continuous or a discontinuous function of P? of /2? of r? 8. The present work will be concerned for the most part with algebraic functions involving only the first and second powers of the variable, i.e., with algebraic equations of the first and second degree. A review is therefore given of the solution and theory of the quadratic equation, presenting in brief the most important results which will be needed in the Analytic Geometry. The student should become thoroughly familiar with this theory, as well as with the review of the trigonometry which follows it. 9. The quadratic equation. Its solution. The most general equation of the second degree, in one unknown number, may be written in the form ax^ -\- bx -\- c = 0, . . . (1) where «, 5, and c are known numbers. This equation may be solved by the method of " completing the square," which gives x^ -\--x + (--]=[--] --, . . . (i) a \2aJ \2aJ a h . Ir b\^ c .1 i.e., x+^=±^^) -^=±^yb^-^ac, . . . (3) 2 a ^\2aJ a 2a b whence x = ± — - V^^ — 4 ac. . . . (4) 2a 2a If x-^ and x^ are used to denote the roots of eq. (1), they may be written X. = , and x^ = . ... (5) 2a 2a 10 ANALYTIC GEOMETRY [Ch. I. The nature of the roots (5) depends upon the number under the radical sign, i.e., upon 5^ — 4 «(?, giving three cases to be considered, viz.: if P — 4 ac > 0, then the roots are both real and unequal, " a P — 4: ac = 0, then the roots are both real and equal, ' (6) if b^ — 4: ac < 0, then the roots are both imaginary. Thus the chat^acter of the roots of a given quadratic equa- tion may be determined without actually solving the equation, by merely calculating the value of the expression 6^ — 4 ac. This important expression is called the discriminant of the quadratic equation ; when equated to zero it states the co7i- dition that must hold among the coefficients if the equation has equal roots. EXERCISES 1. Show which of the following equalities are identities : (1) a-2-4.r + 4 = 0; (4) {p + qy = p^ ^ ^3 + 3^,^ (p + ^). (2) {s + t){s-t) = s^-t'^', (5) a;2 + 5 a; + 6 = (a; + 3)(a; + 2). a + p 2. Determine, without solving the equation, the nature of the roots of 3a:2 + 8a;+l = 0. Solution. Since 6^ _ 4 qc = 64 — 12 = 52, i.e., is positive, therefore the roots are real and unequal ; again, since a, b, and c are all positive, therefore both roots are negative (cf. eq. (4), Art. 9). 3. Without solving the equation, determine the character of the roots of Sx^ — 3x-\-l=0. 4. Given the equation x^ — dx — 7n(x + 2 x'^ + 4:) = 5x^ + 3. Find the roots. For what values of m are these roots equal ? 5. Determine, without solving, the character of the roots of the equations : (1) 5^2 _ 2 2 + 5 = ; (2) a;2 + 7 = ; (S) dt'^ - t = 19. 9-10.] INTBODUCTION 11 6. Determine the values of m for which the following equations shall have equal roots : (1) a;2 - 2a;(l -f 3 m) + 7 (3 + 2 m)=0) (2) mx"^ + 2 z2 - 2 m = 3 mx - 9 a; + lO ; (3) 4x2 + (l + m)^ + 1 = 0; (4) x2 + (6x- + m)2 = a2. 7. If in the equation 2 ax {ax + nc) + (n^ - 2) c^ = 0, a; is real, show that n lies between — 2 and + 2. 8. If X is real in the equation ;: = a, show that a lies between 1 and — ■^. 9. For what values of c will the following equations have equal roots ? (1) 3a:2 + 4x + c=0; (2) (mx + cy = ^lx; (3) 4^2 + 9(2^;+ c)2 = 36. 10. Solve the equations in examples 2, 3, and 5. 11. Solve the equations : (1) 24-25a;2 = -144; (2) 3 a: - 2 _ 2 a: +1 _^ 12 ^ ^^ ^^ '^^a;-2 a; + 2a;2-4 10. Zero and infinite roots. In tlie following pages it will sometimes be necessary to know the conditions among the coefficients of a quadratic equation that will make one or both of its roots zero, or the conditions that will make one or both of the roots infinitely large. In equations (5) of Art. 9, x-^ and x^^ i.e. the roots of ax^ -\- hx -\- e = 0^ were found ; and it is at once seen that — h -{- V52 — 4 «c ^1 = 7^ 2a -h--\/b^-4:ac -h-^b^-4:ac and that b-VW^4^c_ 2c ... (2) Equations (1) and (2) show that : (1) If a and b remain unchanged while c grows smaller, 12 ANALYTIC GEOMETRY [Ch. I. then x^ grows smaller and x^ gro^YS larger ; and if c = 0,* then X. = 0, while x^ = i "a (2) \i a remains unchanged while c = and 5 = 0, then x^ = and a^g = 0. (3) If 5 and c remain unchanged while a = 0, then x-^= and x^^ becomes infinitely large. (4) If c remains unchanged while « = and 5 = 0, then both x-^ and x^ become infinitely large. (5) If a and c remain unchanged while 5 = 0, then x-, = \ and x^ = —\ The student should translate (1), (2), (3), (4), and (5) into more general terms by reading "the absolute term approaches zero as a limit" instead of "^ = 0," etc. 11. Properties of the quadratic equation. By adding the two roots of a^ ^-hx-V c^^ . . . (1) and also multiplying them together, the relations X. -\- Xo = and x.x^ = - . . . (2) a ^ a are obtained ; or, if equation (1) is written with the coeffi- cient of the term of the second degree reduced to unity, as a^ -[- px -\- q = 0, . . . (3) these relations become ^1 + ^^2 — ~ P ^^^ ^1^2 — 9^ • • • QK) Or, expressed in words : the coefficient of the term of the second degree being unity, the coefficient of the term of * The sign = is read " approaches as a limit." It was introduced by the late Professor Oliver of Cornell University. 10-12.] INTRODUCTION 13 the first degree is the negative of the sum of the roots, while tlie term free from x is the product of the roots. If, therefore, tlie roots of a quadratic equation are not themselves needed, but only theu- sum or product is de- sired, these may be obtained directly from the given equa- tion by inspection. E.g.^ the half sum of the roots of the equation mV + 2(hm - 2 l)x -\-h'^ = x^ + x^ _ _ 2 (5m — 2 Z) _ 2 I — hm 2 2m^ m^ Moreover, if x-^ and x^ are the roots of the equation a;^ + pa: + 5' = 0, • then X — x-^ and x — x^^ are the factors of its first member. For, by equation (4) above, this equation may be written x^ -{- px -\- q = x"^ — (x-^ H- x<^ X -t- x-^x^ = 0, and x^ — (x-^ -\- x^ x + x-^x^ = (2; — x-^) {x — x^^ hence x^ + px + g = (a; — x-^ (x — x^ . Conversely : if a quadratic function can be separated into two factors of the first degree, then the roots can be imme- diately written by inspection. For, if x^ -\- px -{- q = (x — x-^(x — x^^ then the first mem- ber will vanish if, and only if , 2: — a;-^ = ov x — x<^ = ; i.e. x^ -\- px -\- q = if X =x-^^ or x = Xg, hence x^ and x^ are the roots of the equation x^ -\- px -{- q = (cf. Art. 4). 12. The quadratic equation involving two unknowns. One equation involving two unknown numbers cannot be solved uniquely for the values of those numbers which satisfy the equation ; but if there is assigned to either of those num- 14 ANALYTIC GEOMETRY [Ch. I. bers a definite value, then at least one definite and corre- sponding value can be found for the other, so that, this pair of values being substituted for the unknown numbers, the equation will be satisfied. In this way an infinite number of pairs of values, that will satisfy the equation, may be found. If, however, the equation is homogeneous in the two un- knowns, i.e., of the form aa^ -f- bx^ -\- cy^ — 0, then the ratio x : y may be regarded as a single number, and the equation has properties precisely like those discussed in Arts. 9, 10, and 11. To solve a system consisting of two or more independent simultaneous equations, involving as many unknown ele- ments, it is necessary to combine the equations so as to eliminate all but one of the unknown elements, then to solve the resulting equation for that one, and, by means of the roots thus obtained, find the entire system of roots. EXERCISES 1. Given the equation x^ + 3 a: — 4 + m (3 x^ — 4) — 2 mx^ = 0, find the sum of the roots; the product of the roots; also the factors of the first member. 2. Factor the following expressions : (1) a;2-5a: + 4; (3) mx^-3x-}-c; (5) 3 ^o^ - 94 ?f; ^ - 64 ; (2) x^ + 2x-8', (4) ax^ + bxy + cf; (6) U-27y-18y^. 3. Without first solving the equation x^ — 3 X — m (x -\- 2 x'^ -{- 4:) = bx^ -\- S find the sum, and the product, of its roots. For what value of m are its roots equal? For what value of m do both its roots become infinitely- large ? If all the terms are transposed to one member, what are the factors of that member? 4. Without first solving, determine the nature of the roots of the equation (jn — 2) (log xy — (2m + 3) log x — 4:m = 0. [Regard log x as the unknown element.] 12-13.] INTRODUCTION 15 For what values of m are the roots equal? Real? One infinitely- great? Both infinitely gTeat? One zero? Find the factors of the first member of the equation. 5. Find five pairs of numbers that satisfy the equation : (1) x + 3y-7=0; (3) y^ = \Qx; (2) a;2 + 2/2 = 4:; (4) 3x + 6x?/-83/2 + 3 a:2 = 0. 6. Without solving, determine the nature of the roots of the equation : 9x2+ \2xy + 42/2 = 0, 3^2 - wy + 19^2 = 0. 7. Solve the following pairs of simultaneous equations : (1) 3a:- 5y + 2 = 0, and 2a; + 7?/ -4 = 0; (2) 5y + 22 + 3 =0, and7y + 4^ + 2 = 0; (3) y = 3 a; + c = 0, and y^ = ^ x', (4) a:2 + 2/2 = 5^ and y^ = Qx; (5) h^x^ + a^y'^ = a^W; and y = ax -\- h', (6) £--f.L = l, and --^ = 1. ^ Me 9 16 9 8. Determine those values of h for which each of the following pairs of equations will be satisfied by two equal values of y : (1) {a;2 + 2/2 = a2, y = Qx + &}; (2) {y = mx +b, y^ = 4a:}; (3) {82/ + 2 a: = &, 6x2 + ^2 ^ 12}. 9. Determine, for the pairs of equations in Ex. 8, those values of h which will give equal values of a:. TRIGONOMETRIC CONCEPTIONS AND FORMULAS 13. Directed lines. Angles. A line is said to be directed when a distinction is made between the segment from any point A of the line to another point By and the opposite seg- ment from B to A. One of these directions is chosen as positive, or +, and the opposite direction is then negative or — . The angle formed by two intersecting directed straight lines is that relation between the positions of the two lines which is expressed by the amount of rotation about their point of intersection necessary to bring the positive end 16 ANALYTIC GEOMETRY [Ch- I. of the initial side into coincidence with the positive end of the terminal side. The point in wliich the lines in- tersect is called the vertex of the angle. The angle is positive^ or 4- , if the rotation ' from the initial to the ter- minal side is in counter-elockwise direction ; the angle is 7iegative, or — , if the rotation is clockwise. The angle formed by two directed straight lines in space, which do not meet, is equal to the angle between two inter- secting lines, which are respectively parallel to the given lines. For the measurement of angles there are two absolute units : (1) The angular magnitude about a point in a plane, i.e., a complete revolution. One fourth of a complete revolution is called a right angle, g^^ of a right angle is a degree (1°), g^Q- of a degree is a minute (1'), and ^^ of a minute is a second (!"} ; (2) the angle whose subtending circular arc is equal in length to the radius of that arc; this angle is called a radian )T^^j ; it is independent of the length of the radius. o- circumference semi-circumference ., r? n ,i , Since = : = TT, it follows that diameter radius the angle formed by a half rotation, i.e., 180°, is ir radians; i.e., 180° = TT^'^^ = (-yj approximately; also l^'') = i?^ = 57° 17' 44.8" approximately. TT (r) A right angle is 90° or f — j When there is no danger of being misunderstood, the index TT -,. . ., , -1 TT (r) is omitted, and — radians is written simply as — , and ... if" 13-14.] INTRODUCTION 17 14. Trigonometric ratios. If from any point P in the ter- minal side of an angle ^, at a distance r from the vertex, a perpendicular MP is drawn to the initial side meeting it in X M Fig. 3.^ M X V ^ Fig. 2v^ ilt/, and if MP be represented by y and VM by x^ then, by general agreement, y is + if MP makes a positive right angle with the initial line, and — if this right angle is negative ; similarly, a; is + if VM extends in the positive direction of the initial line, and — if it extends in the opposite direction. The three numbers r, x^ and y form with each other six ratios ; these ratios, moreover, depend for their value solely upon the size of the angle ^, and not at all upon the value of r. These six ratios are known as the trigonometric ratios or functions of the angle ^, and are named as follows : sine e = y tangent = secant 6 = X X X cosine = -. cotansfent =-. r ^ y cosecant 6 = 1/ The abbreviated symbols for these functions are sin 0, cos 0, tan 6, cot 6, sec ^, and esc ^, respectively. The func- tions are not all independent, but are connected by the fol- lowing relations : (1) sin • CSC = 1, (2) cos ^ • sec (9 = 1, (3) tan ^ . cot ^ = 1, (4) tan = sin 6 : cos ^, TAN. AN. GEOM. 2 (5) cot = cos 6 : sin ^, (6) sin2 -f- cos2 (9 = 1, (7) tan2 ^ + 1 = sec2 6>, (8) cot2 6 + 1 = csc2 6. 18 ANALYTIC GEOMETRY [Ch. L By means of these eight relations all the trigonometric functions of any angle may be expressed in terms of any given function. -^.^■, suppose the sine of an angle is given, and the tangent of this angle, in terms of the sine, is wanted: by (4), tan = ^, -^ ^ ^ cos^ hence tan 6 = and by (6), cos = Vl — sin^ 0, sill 6 Vl-sin2(9 If the numerical value of sin^ is given, this last formula gives the corresponding numerical value of tan^; e.g., if sin ^ = #, then _3 q ^ tan ^ = s = ± T • 15. Functions of related angles. Based upon the defini- tions of the trigonometric functions the following relations are readily established. If 6 is any plane angle, then* (1) sin ( — ^) = — sin 0, cos ( — ^) = + cos 6, tan ( — ^) = — tan 6, esc ( — ^) = — esc 6, sec ( — ^) = -f- sec 6, cot ( — ^) = — cot 6 ; (2) sin (tt ± ^) = T sin 6, cos (tt ± ^) = — cos 0, tan (tt ±0)= ± tan 6, esc (tt ± ^) = T esc 0, sec (tt ±6}= — sec 6, cot (tt ± ^) = ± cot ^ ; (3) sin fE±e^= + cos 6^, cos/| ±0\=T sin ^, tan^l^ T 6>')= T cot 0, ^^^(^ ±o)= + sec 0, secf'^TO)^ T CSC (9, cot^^ ±0)=t tan^. * The student should thoroughly familiarize himself with these formulas, and those of Art. 16, as well as with the derivation of each. 14-16.] INTRODUCTION 19 16. Other important formulas. If ^^ and 6^ ^^^ ^^J two plane angles, then sin (^^ ± ^2) = sin 6^ cos ^2 i c^s ^^ sin 0^, cos (^j ± ^2) = cos 0-^ cos ^2 -^ sin ^^ sin ^g^ 4. ra ^ a \ "tan ^^ ± tan 6^ tan (^. ± a'o) = z ^-7^ ^• ^ ^ ^^ 1 T tan 6>i tan ^2 If 6 is any plane angle, then sin 2 ^ = 2 sin cos ^, cos 2 ^ = cos2 6^ - sin2 6' = 1 - 2 ^\^2 0^2 cos2 ^ - 1, J. oa 2 tan ^ tan 26 = -— ^ 1 - tan2 e sin - = Vj(l — cos ^), Li COS - = V^(l + cos ^), Vi . 9 _ ^\\ — cos ^ 1 — cos ^ sin 2 ^ 1 + cos ^ sin ^ 1 4- cos ^ If a^ 5, and i, l^g, 6'3, ^4, and 6^ be the angles that PQ, QR, RS, ST, and PT respectively make with the line AB, then the projection of the broken line upon AB may also be expressed thus : proj. PQRST upon AB = MN+ NK^ KL + LH= MH = PQ cos (9i + QR cos 6^ + i^^S^cos 6^ + STcos 0^ = PT cos e^. The projections of two parallel segments of equal length upon any given line in space are equal. It therefore fol- lows that : (1) The projection of a segment of a line upon any straight 17.] INTRODUCTION 23 line in space equals the product of its length by the cosine of the angle between the two lines. (2) The projection of any broken line in space upon any straight line equals the projection, upon the same line, of the straight line which joins the extremities of the broken line. EXERCISES 1. Two lines of lengths 3 and 7 respectively meet at an angle -; find the projection of each upon the other. 2. The center of an equilateral triangle, of side 5, is joined by a straight line to a vertex ; find the projection of this joining line upon each side of the triangle. 3. A rectangle has its sides respectively 4 and 6 ; find their projec- tions upon a diagonal. 4. Find the length of the projection of each edge of a cube upon a chosen diagonal. 5. A given line AB makes an angle of 30° with the line MN, and BC is perpendicular to AB and of length 15; find the projection of BC upon MN. Solve this problem if the given angle be a instead of 30°. 6. Two lines in space, of length a and b respectively, make an angle Q) with each other ; find the projection of b upon a line that is perpen- dicular to a. 7. Project the perimeter of a square upon one of its diagonals. CHAPTER II GEOMETRIC CONCEPTIONS. THE POINT I. COORDINATE SYSTEMS 18. Coordinates of a point. Position, like magnitude, is relative, and can be given for a geometric figure only by reference to some fixed geometric figures (planes, lines, or points) which are regarded as known, just as magnitude can be given only by reference to some standard magni- tudes which are taken as units of measurement. The posi- tion of the city of New York, for example, when given by its latitude and longitude, is referred to the equator and the meridian of Greenwich, ^- the position of these two lines being known, that of New York is also known. So also the position of Baltimore may be given by its distance and direction from Washington ; while a particular point in a room may be located by its distances from the floor and two adjacent walls. If, as in the last illustration, a point is to be fixed in space, then three magnitudes must be known, referring to three fixed positions. If, on the other hand, the point is on a known surface, as New York or Baltimore on the surface of the earth, then only tioo magnitudes need be known, referring to two fixed positions on that surface ; while if the point is on a known line, onl}^ one magnitude, referring to one fixed position on that line, is needed to fix its position. These various magnitudes which serve to fix the position 24 Ch. II. 18-20.] GEOMETRIC CONCEPTIONS 25 of a point, — in space, on a surface, or on a line, — are called the coordinates of the point. 19. Analytic Geometry. Coordinates may be represented by algebraic numbers ; the relations of the various points, and tlie properties of the various geometric figures which are formed by those points, can be studied through the corre- sponding relations of these algebraic numbers, or coordinates, expressed in the form of algebraic equations. This fact is the basis of analytic, or algebraic, geometry, the main object of which is the study of geometric properties by algebraic methods. Analytic geometry may be conveniently divided into two parts : Plane Analytic Geometry, which treats only of figures in a given plane surface ; and Solid Analytic Geometry, which treats of space figures, and includes Plane Analytic G-eometry as a special case. The plane analytic geometry, being the simpler, will be studied first, in Part I of tliis book, and Part II will be devoted to the study of the solid analytic geometry. In this first part of the subject it will therefore be understood tliat the work is restricted to a given plane surface. Two systems of coordinates will be used, the Cartesian and the Polar. They are explained in the next few articles. 20. Positive and negative coordinates. If a point lies in a given directed straiglit line, its position with reference to a fixed point of that line is com- pletely determined by one coor- x' P i P x dinate. E.g., let X' OX he a L--3----l-^-3--^ ^ ' Fig. 6. given directed straight line, and let distances from toward X be regarded as positive, then distances from toward X' are negative. A point P, «# 26 ANALYTIC GEOMETRY [Ch. IL in this line and 3 units from toward X may be designated by + 3, where the sign + gives the direction of the point, and the number 3 its distance, from 0. Under tliese circum- stances the point P' lying 3 units on the other side of would be designated by — 3. In the same way there corresponds to every real number, positive or negative, a definite point of this directed straight line; the numbers are called the coordinates of the points; and 0, from which the distances are measured, is called the origin of coordinates. 21. Cartesian coordinates of points in a plane. Suppose two directed straight lines X' OX and Y' OY are given, fixed in the plane and intersecting in the point 0, These two given lines are called the coordinate axes, X' OX being the a;-axis, and Y'OY being the ?/-axis ; their point of inter- section is the origin of coordinates. Any other two lines, parallel respectively to these fixed lines, and at known distances from them, will intersect in one and but one point P, whose position is thus definitely fixed. If these lines through P meet the axes in M and L respectively, then the directed distances LP and MP^ measured parallel respectively to the axes^ are the Cartesian coordinates of the point P. The distance LP^ or its equal OM, is the abscissa of P, and is usually represented by x^ while JfP, or its equal OL, is the ordinate of P, and is usually represented by y. The point P is designated by the symbol (a:, y), — often written P = (x^y')^ — the abscissa always being written first, then a comma, then the ordinate, and both letters being Fig. 7.— Y II. L I. ! X M ' III. IV. 20-22.] GEOMETRIC CONCEPTIONS 27 inclosed in a parenthesis. Thus the point (4, 5) is the point for which OM = 4 and MP = 5 ; while the point (- 3, 2) has 0M= - 3 and MP = 2. 22. Rectangular coordinates. The simplest and most com- mon form of Cartesian coordinate axes is that in which the angle XOY is a positive right angle ; the abscissa (a:) of a point is, in this case, its perpen- dicular distance from the y-axis, and its ordinate (^) is its perpen- dicular distance from the a;-axis. This way of locating the points of a plane is known as the rec- tangular system of coordinates. fig.7.- The axes divide the entire plane into four parts called quad- rants, which are usually designated as first (I), second (11), third (III), and fourth (IV), in the order of rotation from the positive end of the 2:-axis toward the positive end of the y-axis, as indicated in the accompanying figure. These quadrants are distinguished by the signs of the coordinates of the points lying within them, thus : in quadrant I the abscissa (x^ is -H, the ordinate (?/) is + in quadrant II the abscissa (x^ is — , the ordinate (z/) is + in quadrant III the abscissa (a;) is — , the ordinate (?/) is — in quadrant IV the abscissa (2:) is H-, the ordinate (^) is — . Four points having numerically the same coordinates, but lying one in each quadrant, are symmetrical in pairs Avith regard to the origin, even though the axes are not at right angles ; if, however, the axes are rectangular, then these points are symmetrical in pairs, not merely with regard to the origin as before, but also with regard to the axes, and 28 ANALYTIC GEOMETRY [Ch, II. they are severally equidistant from the origin. Because of this greater symmetry rectangular coordinates have many advantages over an oblique system. In the folloiuing pages rectangular coordinates will always be understood unless the contrary is expressly stated, EXERCISES 1. Plot accurately the pomts : (1, 7), (-4, -5) * (0, 3), and ("3, 0). 2. Plot accurately, as vertices of a triangle, the points : (1, 3), (2, 7), and (~4, ~4). Find by measurement the lengths of the sides, and the coordinates of the middle point of each side. 3. Construct the two lines passing through the points (2, -7) and (-2, 7), and (2, 7) and ("2, -7), respectively. What is their point of intersection ? Find the coordinates of the middle point of each line. 4. If the ordinate of a point is 0, where is the point? if its abscissa is 0? if its abscissa is equal to its ordinate? if its abscissa and ordinate are numerically equal but of opposite signs ? 5. Express each of the conditions of Ex. 4 by means of an equation. 6. The base of an equilateral triangle, whose side is 5 inches, coincides with the X-axis ; its ndddle point is at the origin ; what are the coordinates of the vertices? If the axes are chosen so as to coincide with two sides of this triangle, respectively, what are the coordinates of the vertices? 7. A square whose side is 5 inches has its diagonals lying upon the coordinate axes; find the coordinates of its vertices. If a diagonal and an adjacent side are chosen as axes, what are the coordinates of the vertices? of the middle points of the sides? of the center? 8. Find, by similar triangles, the coordinates of the point which bisects the line joining the points (2, 7) and (4, 4). 9. Show that the distance from the origin to the point (a, h) is Va^ + 62. How far from the origin is the point (a, ~h) ? {-a, b) ? (-a, ~b)l (cf. Art. 22.) 10. Prove, by similar triangles, that the points : (2, 3), (1, ~3), and (3, 9) lie on the same straight line. 11. Solve exercises 1 to 4 and 10 if the coordinate axes make an angle of 60°. Also if this angle be 45^ * These minus signs are written high merely to indicate that they are signs of quality and not of operation. 22-23.1 GEOMETRIC CONCEPTIONS 29 23. Polar coordinates. If a fixed point is given in a fixed directed straight line OR, then the position of any point P of the plane will be fully determined by its distance Fig. 8.— ^R OP == p from the fixed point, and by the angle which the line OP makes with the fixed line. The fixed line OR is called the initial line or polar axis, the fixed point the pole of the system, and the polar coordinates of the point P are the radius vector p and the directional or vectorial angle 0. The usual rule of signs applies to the vectorial angle 6, and the radius vector is positive if meas- ured from along the terminal side of the angle 6. The point P is designated by the symbol (p, 6). From what has just been said it is clear that one pair of polar coordinates (i.e., one value of p and one of ^) serve to determine one, and but one, point of the plane. On the other hand, if is restricted to values lying between and 2 TT, then any given point may be designated hjfour different pairs of coordinates. P /-3 240'' / i? R Fig. 9.-^ Fig. 9.-^ >^ ^.^., the polar coordinates (3, 60°) determine the position of the point P, for which OP = 3, and makes an angle of 60° 30 ANALYTIC GEOMETRY [Ch. II. with the initial line 0R\ but the same point may be given equally well by the pairs of coordinates : (~3, 240°), (3, -300°), and ("3,-120°); and so in general. -300>- FiQ.9.- /-3 '120^ Fig. 9.^ R EXERCISES 1. Plot accurately the following points : (2, 20°), [2, -\ ( 7, ^V (477,^), (2,14 0, (-1,-180°), (7,-45°), (-7,135°), (s, ?^), (^'f)' (^'t)' ^^'^°^' ^°^ ^"^'^'^• 2. Construct the triangles whose vertices are: (2,-), (3,—) / 5^\ \ 8/ \ 4 /> (1, -7-)'^ fi^^^ by measurement the lengths of the sides and the coordi- nates of their middle points. 3. The base of an equilateral triangle, whose side is 5 inches, is taken as the polar axis, with the vertex as pole ; find the coordinates of the other two vertices. 4. Write three other pairs of coordinates for each of the points (2, 1); (-3,75°); (5,0°); (0,60°). 5. Where is the point whose radius vector is 7 ? "whose radius vector is —7? whose vectorial angle is 25°? whose vectorial angle is O^*") ? whose vectorial angle is — 180°? 6. Express each of the conditions of Ex. 6 by means of an equation. 7. What is the direction of the line through the points ( 3, — ] and 24. Notation. In the following pages, to secure uniformity and in accordance with Art. 6, a variable point will be desig- 23-26.] GEOMETRIC CONCEPTIONS 31 nated by P, and its coordinates by (x^ y) or (p, ^). If several variable points are under consideration at the same time, tliey will be designated by P, P', P" , P'", •••, and their coordinates by (a;, ?/), (x' , y')^ (x" ^ y'), (x"\ y"'^-> •••, or by (/., ^), (p^ 0% (p", 0^'), (p"', 6'"), .... Fixed points will be designated by P^, Pg, •••, and their coordinates by (^1^ 3/i)' (% «/2)' •••' 01' ^y (/'i' ^i)' (^2^ ^2)' •••• II. ELEMENTARY APPLICATIONS 25. The methods of representing a point in a plane that have been adopted in the previous articles lead at once to several easy applications, such as finding the distance be- tween two points, the area of a triangle, etc. The form of the results will depend upon the particular system of coordi- nates chosen, but the method is the same in each case. Here, as in the more difficult problems that arise later, to gain the full advantage of the analytic method the student should freely use geometric constructions to guide his alge- braic work, but he should, at the same time, see clearly that the method is essentially algebraic. 26. Distance between two points. (1) Polar coordinates. Let OB be the initial line,* the pole, and let P^ = (/o^, 6^} and Pg = (/Og, ^2) ^® ^^^ ^^^ given Fig. 10. Fig. 10.-^ * The demonstration applies to each figure. 32 ANALYTIC GEOMETRY. [Ch. II. fixed points. It is required to find the distance P^Pg — ^ in terms of the given constants /a^, p^, 0^, and 6^. In the triangle OF^P^^ (cf. Art. 16) P^2^ Oi\' + OF^-2 . OF^ ■ OP^ . cos P^OP^. hence ^' = Pi + /^2^ - 2 f)i/>2 cos ((92 - 6'i), ci = Vpi» + P3» - 2piP2 cos (9^ - 0j) [1] (2) Cartesian coordinates ; axes not rectangular. Let OX and O^T be the coordinate axes, meeting at an angle Ya 7 Yl LA B .Q "IP. Fig. 11. 10 M^ M, Fig. 11 A XOY= CO* and let P^ = (x^, y^ and P^ = (x^, y^ be the two given points ; it is required to find the distance P^P^ = d in terms of x^^ x^, ?/j, y^^ and o). Construction : Extend the abscissa L^P^ of the point P^ to meet the ordinate M^P^ of the point P^, in Q ; then in the triangle P^QP^^ (cf. Art 16) F^Pi - F\Q^ -^QP^-2-P,Q- QP,^ . cos P,QP,. Fig. 11«, PJ^ = p;q' + F;q^ -2.P,Q. P,Q . cos P,QP^, Fig. ll^ AA' =QP? + P^ -2'QP^. P,Q . cos Pi§P2, Fig. 11^ ; which gives, for each figure, ^= V(Xi - i»2)3 + (2/1 - 1/3)2+ 2 (iCi - a?3)(i/i - t/3)C0S (O.t * The demonstration applies to each figure. t By examining other possible constructions the student should assure himself of the generality of this formula. 26-27.] GEOMETRIC CONCEPTIONS 33 (3) Hectmigular coordinates. If co = — , i,e. if the coordi- nate axes are rectangular, then cos o) = 0, and the formula for the distance between the two given points becomes i-^3), and 0P2Pi = l P2P1 sin (Oi — O^)- •'• ^= J J/32P3sin(^3-6'2)+/93/3isin(6>i-(93)-/32Pisin((9i-6'2)i, which may also be written A = i \P1P2 sin (^2-^1) + ^2^3 sin (6^3 - ^2) +/03P1 sin((9i-6'3)S. . . . [5] The symmetry* in formulas [4], [4a], and [5] should be carefully noted ; it may be remarked also, that in the appli- cation of these formulas to numerical examples, the resulting areas will be positive or negative according to the relative order in which the vertices are named. * This kind of symmetry is known as cyclic (or circular) symmetry. If 1/^3 the numbers 1,2, and 3 be arranged thus \kJj- , then the subscripts in the •7 first term (in [4a] say) begin with 1 and follow the arrow heads around the circle {i.e. their order is 1, 2, 3), those of the second term begin with 2 and follow the arrow heads (their order is 2, 3, 1), and those of the third term begin with 3 and follow the arrow heads. 29-30.] GEOMETRIC CONCEPTIONS 37 EXERCISES 1. Find the areas of the following triangles: (1) vertices at the points (3, 5), (4, 2), and (1, 3); (2) vertices at the points (7, 3), (4, 6), and (3, -2) ; (3) vertices at the points (11, 9), (0, "2), and (-5, 3). Solve without using the formula, and then verify by substituting in the formula. 2. Prove that the area of the triangle whose vertices are at the points (2, 3), (5, 4), and (-4, 1) is zero,, and hence that these points all lie on the same straight line. 3. Do the points (2, 3), (1, "3), and (3, 9) lie on one straight line? (cf. Ex. 10, p. 28.) 4. Do the points (7, 30°), (0, 0°), and (-11, 210°) lie on one straight line? Solve this by showing that the area of the triangle is zero, and then verify by plotting the figure. 5. Find the area of the triangle ( tt, - j , ( 2 tt, - j , and ( — tt, ^ J . 6. Derive formula [4] when P^ is in quadrant II, P^ in quadrant III, and P3 in quadrant IV. 7. Find the area of the first two triangles in Ex. 1 if the axes make an angle of 60° with each other. 30. To find the coordinates of the point which divides in a given ratio the straight line from one given point to another. Let P^ = (x-^^ y-^ and P^ = (x^, y^ be the two given points, P<^^{x^^ y^ the required point, and let the FiG.U.- FiG. u.- ratio of the parts into which Pg divides P^P^ he m-^ : mcy^ ; ^.e., let PjPg : P3P2 — ^^1 * '^h' I^^^w the ordinates M^P^^ 38 ANALYTIC GEOMETRY [Ch. II, Mc^P^, M^P^, and, through P^ and Pg, draw lines parallel to OX, meeting M^P^ and M^P^ in R and Q respectively. To find OM^ = Xq and M^P^ = t/^ in terms of x^, x^, y^, y^, TTij, and m^. The triangles P^RP^ and P^QP^ are similar; P,R _ RP, _ P,P, therefore But P^Q QP.^ P^P^ P,P m. P^P^ ^«2 X. 3' and P^R = ^3 — ^p P%Q = ^2 ~~ I^Pz = 3/3 - Vv QPi = ^2 - ^3- [In Fig. 14 (^)), a;p i/^, y^, and 2/3 are negative.] . theretore — — — — _ — — - ' *^2 *^3 y^ y^ 2 whence a?3 = ^ ^ — and 2/3 = ^ . . . o . The above reasoning applies equally well whatever the value of ft) (the angle made by the coordinate axes), hence formulas [6] hold whether the axes be rectangular or oblique. Formulas [6] were obtained on the implied hypothesis that P3 lies between P^ and P^ ; i.e., that Pg is an inte7mal point of division. If P3 is taken in the line P1P2 produced, and not between P^ and P^, it still forms, with P^ and P^, two segments P1P3 and P^P^^ and P3 may be so taken that, numerically, the ratio of P1P3 : P?,^^ ^^^^J bave any real value whatever ; but the sign of this ratio is negative when Pg is not between P^ and Pg, for, in that case, the segments PjPg and P3P2 have opposite directions. Hence, to find the coordinates of that point which divides a line externally into segments whose numerical ratio is m-^ : m,^, it is only 30.] GEOMETRIC CONCEPTIONS 39 necessary to prefix the minus sign to either one of the two numbers m^ or m^ in formulas [6]. These formulas then become ^ m^x^ - m^x^ ^ rn^y^. -^2^1 . . . [-71 Cor. If -P3 be the middle point of P^P^^ then m-^ = and formulas [6] become w 2' a^s - — 2~' ^3 - ~^~ ; . . . [8] ^.g., the abscissa of the middle point of the line joining two given points is half the sum of the abscissas of those points, and the ordinate is half the sum of their ordinates. The remarks in Art. 28 are well illustrated by formulas [4] to [8]; EXERCISES 1. By means of an appropriate figure, derive formulas [7] independ- ently of [6]. 2. The point Pg = (2, 3) is one third of the distance from the point P^={-1, 4) to the point Po={x2, 2/2) ; to find the coordinates of Pg- Here P^ and P3 are given, with a;^ = — 1, ?/i = 4, x^ = 2, y^ = 3, also Tn■^^ = 1, and 7/12 = 2 ; therefore, from [6], g_ ^2 + 2(-l) ^^^ 3_ y. + 2(4) which give x^ = S and ^2 = 1 1 therefore the required point Pg is (8, 1). 3. Find the points of trisection of the line joining (1, -2) to (3, 4). 4. Find the point which divides the line from (1, 3) to (-2, 4) externally into segments whose numerical ratio is 3 : 4. Here x^ = 1, y^ = 3, Xg = — 2, ^3 = 4, m^ = 3, and m^ = 4, but the point of division being an external one, the two segments are oppo- sitely directed ; therefore one of the numbers 3 or 4, say 4, must have the minus sign prefixed to it. Substituting these values in [6], 3(l)-4(-2) ^_ ^^^ 3(3)-4(4)^ 3 3 _ ^ ys 3 _ ^ « , the required division point is, therefore, Pg = (~11, 7). 40 ANALYTIC GEOMETRY [Ch. II. The same result would have been obtained had m.^ — 3, instead of TWg = 4, been given the minus sign ; or, again, formulas [7] could have been employed to solve this jDi'oblem. 5. Solve Ex. 4 directly from a figure, without using either [6] or [7]. 6. Find the points which divide the line from (1, 5) to (2, 7) inter- nally and externally into segments which are in the ratio 2 : 3. 7. A line AB is produced to C, so that BC = ^AB; if the points A and B have the coordinates (5, 6) and (7, 2), respectively, what are the coordinates of C ? 8. Prove, by means of Art. 30, that the median lines of a triangle meet in a point, which is for each median the point of trisection nearest the side of the triangle. 31. Fundamental problems of analytic geometry. The elementary applications already considered have indicated how algebra may be applied to the solution of geometric problems. Points in a plane have been identified with pairs of numbers, — the coordinates of those points, — and it has been seen that definite relations between such points corre- spond to definite Telations between their coordinates. It will be found also that the relation between points, which consists in their lying on a definite curve, corre- sponds to the relation between their coordinates, which consists in their satisfying a definite equation. From this fact arise the two fundamental problems of analytic geom- etry : I. Given an equation^ to find the corresponding geometric curve, or locus. II. Griven a geometric curve, to find the corresponding equation. When this relation between a curve and its equation has been studied, then a third problem arises : III. To find the properties of the curve from those of its equation. ,. 30-31.] GEOMETRIC CONCEPTIONS 41 The first two problems will be treated in the two succeed- ing chapters, while the remaining chapters of Part I will be concerned chiefly with the third problem. In this appli- cation of analytic methods, however, only algebraic equa- tions of the first and second degrees will for the most part be considered. In Chapter XIII is given a brief study of other important equations and curves. EXAMPLES ON CHAPTER II 1. Find the area of the quadrilateral whose vertices are the points (1, 0), (3, I), (-1, 16), and (-4, 2). Draw the figure. 2. Find the lengths of the sides and the altitude of the isosceles triangle (1, 5), (5, 1), (-9, ~9). Find the area by two different methods, so that the results will each be a check on the other. 3. Find the coordinates of the point that divides the line from (2, 3) to (~1, ~6) in the ratio 3:4; in the ratio 2 : -3 ; in the ratio 3 : ~2. Draw each figure. 4. One extremity of a straight line is at the point (~3, 4), and the line is divided by the point (1, 6) in the ratio 2:3; find the other ex- tremity of the line. 5. The line from (-6, ~2) to (3, —1) is divided in the ratio 4:5; find the distance of the point of division from the point (~4, 6). 6. Find the area and also the perimeter of the triangle whose vertices are the points (3, 60°), (5, 120°), and (8, 30°). 7. Show analytically that the figure formed by joining the middle points of the sides of any quadrilateral is a parallelogram. 8. Show that the points (1, 3), (2, V6), and (2, —VQ) are equidis- tant from the origin. 9. Show that the points (1, 1), (— 1, —1), and ("Vs, -V3) form an equilateral triangle. Find the slopes of its sides. 10. Prove analytically that the diagonals of a rectangle are equal. 11. Show that the points (0, -1), (2, 1), (0, 3), and (-2, 1) are the vertices of a square. 42 ANALYTIC GEOMETRY [Ch. II. 31. 12. Express by an equation that the point (h, k) is equidistant from (-1, 1) and (1, 2); from (1, 2) and (1, -2). Then show that the point (I, 0) is equidistant from (-1, 1), (1, 2), and (1, -2). 13. Prove analytically that the middle point of the hypotenuse of a right triangle is equidistant from the three vertices. 14. Three vertices of a parallelogram are (1,2), (-5,-3), and (7, -6) ; what is the fourth vertex ? 15. The center of gravity of a triangle is at the point in which the medians intersect. Find the center of gravity of the triangle whose vertices are (2, 3), (4, -5), and (3, -6). (cf. Ex. 8, p. 40.) 16. The line from (x^ y^ to {x^, y^ is divided into five equal parts ; find the points of division. 17. Prove analytically that the two straight lines which join the middle points of the opposite sides of a quadrilateral mutually bisect each other. 18. Prove that (1, 5) is on the line joining the points (0, 2) and (2, 8), and is equidistant from them. 19. If the angle between the axes is 30°, find the perimeter of the triangle whose vertices are (2, 2), (-7, -1), and (-1, 1). Plot the figure. 20. Show analytically that the line joining the middle points of two sides of a triangle is half the length of the third side. 21. A point is 7 units distant from the origin and is equidistant from the points (2, 1) and (-2, -1) ; find its coordinates. 22. Prove that the points (a, 6 + c), {h, c + a), and (c, «-}-&) lie on the same straight line. (cf. Ex. 2, p. 37.) CHAPTER III THE LOCUS OF AN EQUATION 32. The locus of an equation. A pair of numbers x^ y is represented geometrically by a point in a plane. If these two numbers (x^ y) are variables, but connected by an equa- tion, then this equation can, in general, be satisfied by an infinite number of pairs of values of x and ?/, and each pair may be represented by a point. These points will not, however, be scattered indiscriminately over the plane, but will all lie in a definite curve, whose form depends only upon the nature of the equation under consideration ; and this curve will contain no points except those whose co- ordinates are pairs of values which when substituted for X and ?/, satisfy the given equation. This curve is called the locus or graph of the equation ; and the first funda- mental problem of analytic geometry is to find, for a given equation, its graph or locus. 33. Illustrative examples : Cartesian coordinates. (1) Given the equation x -{• o = 0, to find its locus. This equation is satisfied by the pairs of values x^= — o, y^ = 2; a^g = — 5, 3/2 = 8; arg = — 5, 2/3 = — 2 ; etc., that is, by every pair of values for which x = — o. Such points as P,= (x„y,) = i-D,2), Po=(x,,, 2/2) = (-5, 3), ^3 = (^3' 3/3) = (r^^y ~3)» etc., all lie on the line MN, parallel to the y-axis, and at the distance 5 on the negative side of it, — this line extending indefinitely in both direc- 43 44 ANALYTIC GEOMETRY [Ch. III. tions. Moreover, each point of MN has for its abscissa -o, hence the coordinates of each of its points satisfy the equation x + 5 == 0. In the chosen system of coordi- N N C. i? A Pa M O D B Fig. 15. M nates, the line MN is called the locus of this equation. Similarly, the equation a; — 5 = is satisfied by any pair of values of which x is 5, such as (5, 2), (5, 3), (5, 4), etc. ; all the corresponding points lie on a straight line M'N', parallel to the ?/-axis, at the distance 5 from it, and on its positive side ; i.e., M'N' is the locus of the equa- tion X — 5 = 0. (2) Given the equations y ± 3 = 0, to find their loci By the same reasoning as in (1) it may be shown that the locus of the equation ?/ -f 3 = is the straight line AB, parallel to the x-axis, situated at the distance 3 from it, and on its negative side. Also that the locus of the equation ?/ — 3 = is CD, a line parallel to the a:-axis, at the distance 3 from it, and on its positive side. More generally, it is evident that in Cartesian coordinates (rectangular or oUique), an equation of the first degree, and containing hut one variable^ represents a straight line parallel to one of the coordinate axes. (3) Given the equation 3 a; — 2 ?/ + 12 = 0, to find its locus. In this equation both the variables appear. By assigning any definite value to either one of the variables, and solving the equation for the other, a pair of values that will satisfy the equation is ob- tained. Thus the following pairs of values are found : ^1 = 0, ?/i = 6 a^2 ^^ J- ? y 2 ^^ 2 arg = 2, ?/3 = 9 3^4 = ^' 2/4 = 10^ + CO, ?/ =+ CO ^5 = - 1» y.5 = 4^ Xg = - 2, ?/^, = 3 ^r = - ^> 2/r = H 2^8 = - '^' ^8=0 a: = — CO, V = y =- CO Plotting the corresponding points Pj, P„ P„ P, ... , where P^ = {x„ y,) = (0, 6), P.-> = ix^,y^^ = {\, 7i), etc., they are all found to lie on the straight line EF, which is the locus of the equation 3 a; — 2 ?/ -f 12 = 0. Fia.16. 33.] THE LOCUS OF AN EQUATION 45 In Chap. V, it will be shown that, in Cartesian coordinates, an equar tion of the first degree in two variables always represents a straight line. (4) Given the equation ?/2 = 4 x, to find its locus. This equation is satisfied by each of the following pairs of values, found as in (3) above : ^'i = 0, ?/i = ^2 = 1' 2/2 = + 2 ^3 = 1' 2/3 = - 2_ x^ = 2, 3/4 = 2 ^/2 = 2.8, approximately Xg = 2, 2/5 = — 2 V2= —2.8, approximately ^6 = ^5 2/6 = + 4 x = + oo, y = ±. (X) and for any negative value of x the corre- sponding value of y is imaginary. The corresponding points are : ' Pi = (0, 0), P2 = (1.2), P3 = (l,-2), etc. FiG.17. P/ All these points are found to lie on the curve as plotted in Fig. 17. This curve is called a paraLola, and will be studied in a later chapter. The parabola is one of the curves obtained by the intersection of a circular cone and a plane, (cf. Appendix, Note D.) It will be shown in Chap. XII that in Cartesian coordinates, the locus of any alge- braic equation in two variables and of the second degree is a "conic sec- tion." (5) Given the equation, y = 25 log x, to find its locus. A table of logarithms shows that this equation is satisfied by the following pairs of values : x^ = 0, 2/1=-^ xj = 6, 2/7 = 19-4 X2 = 1) ^2 = xs = 7, 2/8 =21.1 Xg = 2, 2/3 = 7.5 x^ = 10, 2/9 =25 x^ = 3, 2/4 = 11-9 -^10 = lOj 2/10 = 29.4 x, = 4, 2/5 = 15 X,, = 20, 2/11 = 32.5 Xn = 5, y. = 17.5 etc. etc. The corresponding points are : P, = (0, - CO), P, = (h 0), P3 = (2, 7.5), etc. ; and the locus of the above equa- tion is approximately given by the curve drawn through these points as shown in Fig. 18. 46 ANALYTIC GEOMETRY [Ch. III. (6) Given the equation y = tan x, to Jind its locus. By means of a table of " natural " tangents it is seen that this equation is satisfied by the following pairs of values of x and y : Dkgrees Eadians x^ = = 0.00 yi =0 X, =10 = 0.17 y, = 0.18 X, =20 = 0.35 y, =0.36 x^ =30 = 0.52 y^ = 0.58 ^5 = 40 = 0.70 y, = 0.84 X, =50 = 0.87 3/6 - 1-19 Xj =60 = 1.05 yr = 1-73 Xg = 70 = 1.22 yg =2.75 Xg =80 = 1.40 2^9 =5.67 X,, = 90 = 1.57 2/io = ^ X,, = - 10 = - 0.17 ^11 = - 0-18 x^^ = - 20 = - 0.35 3/i2 = - 0-36 a:j3 = - 30 = - 0.52 3/13 = - 0.58 etc. etc. etc. The corresponding points are : Pi = (0, 0), P^ = (O.U, 0.18), P3 = (0.3.5, 0.36), etc., and the locus is approximately as shown in Fig. 19. Fig. 19. X 34. Loci by polar coordinates. Analogous results are obtained for a System of polar coordinates, as will be best seen from an example. Given the equation p = 4:Cos 0, to Jind its locus. 33-35.] THE LOCUS OF AN EQUATION 47 This equation is satisfied by the following pairs of values, found as in Art. 33 (3) and (4) : ' 0, = S0° p2 = 2^3 = 3.46 + ^, = 60^ p3 = 2 0^ = 4:5° p4 = 2V2 = 2.8 + ^5 = 90° p, = 6*6 = -30° p6 = 3.46 + ^^ = - 60° p^ = 2 ^3 =---45° p, = 2.8-{- ■ 0, = -90° p, = etc. etc. The corresponding points are : Pi=(4, 0°); P2=(3.46 + , 30°); P, = (2, 60°); P,= C2.S + , 45°); P5=P9= the pole O=(0, ±90°); Pg=(3.46 + , -30"); Pj = (2, -60°); etc. All these points are found to lie on the circumference of a circle whose radius is 2, the pole being on the circumference, and the polar axis being a diameter. This circle is the locus of the equation p = 4 cos 0. EXERCISES Plot the loci of the following equations : 1. x = 0. 7. x2 + y^ = 4. 13. ^2 + ^2 3^ 9, 2. y = 0. 8. a; + y = 4. 14. u'^+v=0. 3. hx = Q. 9. X — y = 0. 15. s = 16 t\ 4. 3a; = 7. 10. r^1_y1 = 4. 16.^ + ^ = 1. 2 3 5. 2y + b = 0. 11. 2 a:2 + 2/2 = 4. 17. p = 3. 6. x + y = 0. 12. V = 32 1. 18. p cos ((9 -40°) = 19. y = - xK 5. 35. The locus of an equation. By the process illustrated above, of constructing a curve from its equation, the first conception of a locus is obtained, viz. : (1) The locus of an equation co7itainin(j two variables is the line^ or set of lines^ which coiitains all the points tvhose coordinates satisfy/ the given equatio7i^ aiid ivhich contaijis no other points. It is the place ivhere all the points, and 48 ANALYTIC GEOMETRY [Ch. III. only those points, are found whose coordinates satisfy the given equation. A second conception of the locus of an equation comes directly from this one, for the line or set of lines may be regarded as the path traced by a point which moves along it. The path of the moving point is determined by the condition that its coordinates for every position through which it passes must satisfy the given equation. Thus the line EF (the locus of eq. (3), Art. 33) may be regarded as the path traced by the point P, which moves so that its coordinates (a;, ?/) always satisfy the equation 3a;-2^ + 12 = 0. Thus arises a second conception of a locus, viz. : (2) The locus of an equation is the path traced hy a point which moves so that its coordinates always satisfy the given equation. In either conception of a locus, the essential condition that a point shall lie on the locus of a given equation is, that the coordinates of the point ivhen substituted respectively for the variables of the equation^ shall satisfy the equation; and in order that a curve may be the locus of an equa- tion, it is necessary that there be no other points than those of this curve ivhose coordinates satisfy the equation. 36. Classification of loci. The form of a locus depends upon the nature of its equation ; the curve may therefore be classified according to its equation, an algebraic curve being one whose equation is algebraic, and a transcendental curve one whose equation is transcendental. In particular, the degree of an algebraic curve is defined to be the same as the degree of its equation. The following pages are 30-37.J THE LOCUS OF AN EQUATION 49 concerned chiefly with algebraic curves of the first and second degrees. 37. Construction of loci. Discussion of equations. The process of constructing a locus by plotting separate points, and then connecting them by a smooth curve, is only ap- proximate, and is long and tedious. It may often be short- ened by a consideration of the peculiarities of the given equation, such as symmetry, the limiting values of the vari- ables for which both are real, etc. Such considerations will often show the general form and limitations of the curve ; and, taken together, they constitute a discussion of the equa- tion. The points where a locus crosses the coordinate axes are almost always useful ; in drawing the curve, they are given by their distances from the origin along the respective axes. These distances are called the intercepts of the curve. The following examples may serve to illustrate these conceptions. (1) Discussion of the equation 3x — 2y-\-12 = [see (3) Art. 33]. Intercepts : if x = 0, then y = Q; hence the ^/'intercept is 6 (see Fig. 16) ; it y = 0, then x = i] hence the x-intercept is 4. The equation may be written : x = ^ y — 4, which shows that as y increases continuously from to go , a: increases continuously from — 4 to CO ; therefore the locus passes from the point Pg through the point P^, and then recedes indefinitely from both axes in the first quadrant. Writ- ten as above, the equation also shows that as y decreases from to — co , X also decreases from — 4 to — co ; therefore the locus passes from Pg into the third quadrant, receding again indefinitely from both axes. Since for every value of y, x takes but one value (i.e., each value of y corre- sponds to but one point on the curve), therefore the locus consists of a single branch. The proof that the locus of any first-degree equation, in two variables, is a straight line is given in Chap. V. (2) Discussion of the equation y'^ = 4x. [See (4) Art. 33.] Intercepts (see Fig. 17) : if a; = 0, then y = 0, and if ?/ = 0, then x — ; TAN. AN. GEOM. 4 50 ANALYTIC GEOMETRY [Ch. III. hence the locus cuts each axis in one point only, and that point is the origin. The equation may be written in the form ?/ = ± Vi x, which shows that if x be negative y is imaginary ; hence there is no point of this locus on the negative side of the ?/-axis. Again : for each positive value of x there are two real values of y, numerically equal, but opposite in sign ; hence this locus passes through the origin, lies wholly in the first and fourth quadrants, and is symmetri- cal with regard to the rr-axis. The equation shows also that x may have any positive value, however great, and that y increases when x increases ; these facts show that the locus recedes indefinitely from both axes, — that it is an open curve of one branch. It is called a parabola and has the form shown in Fig. 17. (3) Discussion of the equation a;^ + y^ _ q2 Intercepts : if a: = 0, then y = ± a, and iJL y = 0, then x = ±a; hence for each axis there are two intercepts, each of length a, and on opposite sides of the origin ; i.e., four positions of the tracing point are : A=(a, 0), A' = (-a, 0), B=(0, a), and ^' = (0, -a). This equation may also be written y = ± Va^ — x'^, which shows that every value of x gives two corresponding values of y which are numerically equal, but of opposite sign ; the locus is, therefore, symmetrical with regard to the x-axis. It also shows that, corresponding to any value of x numerically greater than a, y is imaginary; the tracing point, therefore, does not move further from the ?/-axis than ± a, i.e., further than the points A and A'. Moreover, as X increases from to a, y remains real and changes gradually from + a to 0, or from —a to ; i.e., the tracing point moves continuously from B to A, or from B' to A. Again, if x decreases from to — a, y remains real and changes con- tinuously from + a to 0, or from — a to ; i.e., the tracing point moves continuously from B to A' or from B to ^'. Similarly, the equation may be written x = ± Va^ _ y'i^ which shows that the curve is also symmetrical with regard to the ?/-axis, and that the tracing point does not move farther than ± « from the a:-axis. From these facts it follows that this locus is a closed curve of only one branch. That it is a circle of radius a, with its center at the origin, will be shown in Chap. VII. 37.] THE LOCUS OF AN EQUATION 61 (4) Discussion of the equation y^ = {x — 2) (x — 3) (a: — 4). Intercepts : if a: = 0, then y is imaginari/ ; if ?/ = 0, then a; = 2, 3, or 4; hence the locus crosses the x-axes at the three points: ^=(2, 0), 5 = (3, 0), and C= (4, 0), and it does not cut the ?/-axis at all. Moreover, since y is imaginary if X is negative, the locus lies wholly on the positive side of the ?/-axis. This locus is symmetrical with regard to the ar-axis; it has no point nearer to the ?/-axis than A ; between A and B it consists of a closed branch ; and it has no real points between B and C, but is again real beyond C. The entire locus consists, then, of a closed oval, and of an open branch which recedes indefinitely from both axes, see Fig. 22. Fig.'^-^. j: (5) Discussion of the equation y = tan x. This equation has already been examined in (6) Art. 33, but in practice it may be much more simply plotted by the following method : Describe a circle with unit radius; draw the diameter ADC, and the lines OB^, OB^, OB., •", meeting the tangent AT in the points T^, T^, ^s^"-; tlien the tangent of the angle AOB^ is M^B^^ : OM^ = AT^: OA (Art. 14), and, since OA =1, its value is graphically rep- resented by A Ty So also tan ^0^2 = -^2^2 ' OM^ = AT,^: 0A= AT^:1, and may be graphically represented by A T^. In the same way, A T., A T^, AT^, ••• are the tangents of the angles AOB^, AOB^, AOBq, •••. Again, since angles at the center of a circle are propor- tional to the arcs intercepted by their sides, A T^, AT^, ••• may be said to be the tangents of the i.e., AT^ = tan AB,, A T^ = tan Therefore the coordinates of the points arcs AB^, ABc^, AB,,- P^=(AB„ AT^), P, = {AB.2, AT^),"- satisfy the given equation, and if a sufficient number of points, whose coordinates are thus determined, be plotted, they will all lie on a curve like that in Fig. 19. 52 ANALYTIC GEOMETRY [Cn. III. From what has just been said it is clear that ?/ = if x = 0, hence the curve goes through the origin ; when x increases continuously from to — , y increases continuously from to oo, but when x increases through — , ?/ passes suddenly from + oo to — oo, and the curve is discontinuous for that value of x. So also when x increases continuously from — to '— -, y increases continuously from — oo through to -f oo, and is again dis- continuous for x = — . The locus consists of an infinite number of such infinite, but continuous branches, separated by the points of discon- tinuity for which x = ±—, x = ± ——, x = ± —, •••. The other trigonometric functions, y = sin x, y = sec x, etc., can all be plotted by a method analogous to that above. EXERCISES Construct and discuss the loci of the following equations : (cf . Ex. 8, p. 8.) 38. The locus of an equation remains unchanged: (a) by any transposition of the terms of the equation ; and (P) by multiplying both members of the equation by any finite con- stant. (a) If in any equation the terms are transposed from one member to the other in any way whatever, the locus of the equation is not changed thereby ; for the coordinates of all the points which satisfied the equation in its original form, and only those coordinates, satisfy it after the transpositions are made. [See Art. 35 (1).] (/3) If both members of an equation are multiplied by any finite constant ^, its locus is not changed thereby. For if the terms of the equation, after the multiplication has been performed, are all transposed to the first member, that mem- ber may be written as the product of the constant k and a 1. a;2 ?/2 _ ^ 3. y — sec X. 7. V = sin u. 4 9 4. x'^ — ?/-= or. 8. a'2 M f = 0. 2. ^ + ^ = 1. 4 9 5. 6. x^-f^ 0. 9. ?/-l_5-i y-2 37-39.] . THE LOCUS OF AN EQUATION 53 factor containing the variables. This product will vanish if, and only if, its second factor vanishes ; but this factor Avill vanish if, and only if, the variables which it contains are the coordinates of points on the locus of the original equation. Hence the coordinates of all points on the locus of the ori- ginal equation, and only those coordinates, satisfy the equation after it has been multiplied by k ; hence the locus remains unchanged if its equation is multiplied by a finite constant. 39. Points of intersection of two loci. Since the points of intersection of two loci are points on each locus, therefore the coordinates of these points must satisfy each of the two equations ; moreover, the coordinates of no other points can satisfy both equations. Hence, to find the coordinates of the points of intersection of two curves, it is only necessary to regard their equations as simultaneous and solve for the coordinates. E.g., Find the coordinates of the points of intersection, P^ and Pg' ^^ the loci oi X — 2 y = 0, and y'^ = x. The point of intersection P^= (i\, y^) is on both curves, .'. Xj — 2 y^ = 0, Siud y^ = x^* Solving these tv^o equations, x^ = 0, or 4, and ?/j = 0, or 2 ; i.e., Pi= (4, 2) and P^= (0, 0) are two points, the coordinates of which satisfy each of the two given equa- tions ; therefore they are the points of intersection of the loci of these equations. EXERCISES Find the points of intersection of the following pairs of curves : ^7x-Uy-\-l=0, ^ ^x-\-y = 3. Fig. 31. 1. Ix +y~2 = 0. 2. x-y * If. X and y are regarded as the coordinates of the point of intersection, the subscripts may be omitted here. 54 ANALYTIC GEOMETRY [Ch. III. (y=Sx + 2, ^x + y = 2a, ^ (2y-5x = 0, 9 (^'+y' = 16, ( a;2 - 2^2 _ 5, ix ^y — 1. a:2+2/2 = 9, 10. 1^'=^^' j a;2 a:2 + 6 2'^ + ^^ = 0. •'^ ~ ' ^ (p = 9 COS (45°-^), (?/2 = 4;9^, 12. ^ /7r^^\ , ^' iy-x = 0. lpco.[- + 0) = l. 13. Trace carefully the above loci; by measurement, find the coordi- nates of the points in which each pair intersect ; and compare these results with those already obtained by computation. 40. Product of two or more equations. Given two or more equations with their second members zero ; * the product of their first members^ equated to zero^ has for its locus the combined loci of the given equations. This follows at once from the fundamental relation be- tween an equation and its locus (see Art. 35 (1)), for the new equation is satisfied by the coordinates of those points which make one of its factors zero, but it is satisfied by the coordinates of no other points; ^.6., this new equation is satisfied by the coordinates of points that lie on one or another of the loci of the given equations. The following example illustrates this principle in the case of two given equations. Let the given equations be : x^y z=0 . . . (1) and x-y = . . . (2) * If equations whose second members are not zero are multiplied together, member by member, the resulting equation is not satisfied by any points of the loci of the given equations except those in which they intersect each other ; the new equation therefore represents a locus through the points of intersection of the loci of the given equations. 40.] THE LOCUS OF AN EQUATION 55 Y >? / Pa \ / • \ y X \ \ ^ \o Fig. 35. Equation (1) represents the straight line CZ), and equation (2) the line AB^ — bisecting re- spectively the angles between the axes. It is to be shown that the equation ix + y-){x~y-)=^ . . ,. (3) (or, what is the same, a:^— ?/^= 0), formed from equations (1) and (2), has for its locus both these lines. Proof. If P^ = (iCj, y-^ is any point on CD, then its co- ordinates satisfy equation (1), hence x^-\- yi = 0, and there- fore (x^ + ^j) (x^ — y^ = ; which shows that P^ is a point of the locus of equation (3). But since P-^ was any point of CP, therefore the coordinates of every point on QP satisfy equation (3); i.e., all points of CP belong to the locus of equation (3). In the same way it is shown that AP belongs to the locus of equation (3). Moreover, if P^^{x2^, y^ be any point not on AP nor on CP, then ^3 4- ^3 ^ 0, and x^^ — y^,^^ 0, hence i.e., Pg does not belong to the locus of equation (3). Hence the locus of equation (3) contains the loci of equa- tions (1) and (2), but contains no other points. The above theorem may be stated briefly thus : if u, v, w, etc., be any functions of two variables, then the equation uvw ' ••• =0 has for its locus the combined loci of the equations u = 0, v = 0, w = 0, etc. Note. When possible, factoring the first member of an equation, whose second member is zero, simplifies the work of finding the locus of the given equation. 56 ANALYTIC GEOMETRY [Ch. III. EXERCISES What loci are represented by the following equations V 1. xy = 0. 2. ^-^z=:0. 3. 'dx^ + 2xy-7x = 0. ^ 4 9 4. 5xy-2-2x''y = 0. 5. x2-2a; + l = 0. 6. (xHy^-4){y^-^x)=0. 41. Locus represented by the sum of two equations. Sup- pose the equations 2 ^ - 2) = . . . (1), and j/2 - a: = . . . (2) are given. Their loci are respectively AB and DP^P^C (Art. 39), and it is required to find the locus of their sum ; ^.e., of 2 ?/ — X + ?/^ — rr = 0, or, what is the same thing, of ./2+27/-2:r = . . . ^3) The locus of this last equa- tion passes through all the points in which AB and DP^P^ C intersect each other. For let P-^=(x-^^ y^ be one of these points, then since P^ lies on AB^ its coordinates satisfy equation (1); z.e., 2y^-x, = 0; . . . (4) and since P^ lies on DP^P^C, its coordinates satisfy equa- tions (2); z.e., ^i^-^i = ^5 . . . O'^) therefore, by adding equations (4) and (5), . ^,2 + 2^1 -2:^1 = 0. ... (6) This last equation proves (Art. 35 (1)) that P^=(x^, y^) is on the locus of equation (3); ^.e., the locus of equation (3) passes through P-^ = (^x^, y-^. Similar reasoning would show that the locus of equation Fig. 26. 41.] THE LOCUS OF AN EQUATION 57 (3) passes through every other point in which the loci of equations (1) and (2) intersect each other. In precisely the same way it may be proved generally that the locus of the sum of tioo equations passes through all the points in ivhich the loci of the two given equations intersect each other. If either of the given equations (1) or (2) had been multi- plied by any constant factor before adding, the above reason- ing would still have led to the same conclusion ; in fact, this theorem may be briefly, and more generally, stated thus : if u and v are any functions of the two variables x and y, and k is any constant., then the locus of u -\-hv = passes through every point of intersection of the loci of u = and V = 0. For, let the locus of the equation w = be the curve ABC^ the locus of v = be the curve BEF^ and let Pi = (xi^ yi) be any one of the points in which these curves intersect each other. Then the equation u -{- kv = is satisfied by the coordi- nates of the point Pi = (^n yi)-> because if these coordinates be substituted for x and y in the functions u and V they must make both these functions separately equal to zero. Therefore the locus of u -\- kv — passes through every point in which the loci oi u = and v = intersect each other. Fig. 37 58 ANALYTIC GEOMETRY [Ch. III. EXERCISES 1. Verify Art. 41 by first finding the coordinates of the points of intersection of the loci of equations (1) and (2), and then substituting these coordinates in equation (3). 2. Find the equation of a curve that passes through all the points in which the following pairs of curves intersect : ^""^ lx'' + 2x+i/ = 0. i "^f^^ \y = 2 cos X. j 3. Find the equation of a curve through all the points common to the following pairs of curves : ^"; ly^ = 4:x.\ ^^^ (pcos^:-!.; Note. It is to be observed that the method given in Art. 39, for find- ing the point of intersection of two curves, is an application of the theorem of Art. 41. For the process of solving two simultaneous equa- tions, at least one of which involves two variables, consists in combining them in such a w^ay as to obtain two simple equations, each involving only one variable. Now each of these simple equations represents an elementary locus, — one or more straight lines parallel to the axes, if the coordinates are Cartesian ; circles about the pole, or straight lines through the pole, if the coordinates are polar, — and these elementary loci deter- mine, i.e., pass through, the points of intersection of the original loci. To determine the points of intersection, then, of two loci, the original loci are replaced by simpler ones passing through the same common points. E.g., the points of intersection of the loci of Art. 39, 2y-x = . . . (1), and y^ = x, . . . (2) are given by the equations (y'^-x) -(2y -x)=0 and (2 y - x)^ - ^ (y^ -x) = 0, that is, by y^ — 2 y = 0, and x^ — 4: x = 0, which may be written y(y-2) = . . . (3), x(x-4.) = 0. ... (4) But the locus of equation (3) is a pair of straight lines parallel to the a;-axis, and the locus of equation (4) is a pair of straight lines parallel to the ?/-axis ; and these loci have the same points of intersection as the loci (1) and (2). 41] THE LOCUS OF AN EQUATION 59 EXAMPLES ON CHAPTER III 1. Are the points (3, 9), (4, 6), and (5, 5) on the locus of 3 x-\-2y = 25'i 2. Is the point l^,^\ on the locus of 4 x^ + 9 ^/^ = 2 a^ ? 3. The ordinate of a certain point on the locus of x^ + y'^ = 25 is 4 ; what is its abscissa? What is the ordinate if the abscissa is a'^? Find by the method of Art. 39 where the following loci cut the axes of X and y. 4. y = (x-2)(x-S). 5. IQ x^ + 9 y^ = lU. 6. a;2 + 6 a; + 2/2 = 4?/ 4- 3. Find by the method of Art. 39 where the following loci cut the polar axis (or initial line). 7. p = 4sin2^. 8. p2 ^ a^ cos 2 ^. 9. The two loci — = 1, — 1-^ = 1 intersect in four points; find 4 9 4 9 F » the lengths of the sides and of the diagonals of the quadrilateral formed by these points. 10. A triangle is formed by the points of intersection of the loci of x -{■ y = a^ a: — 2 2/ = 4a, and y — x -\-1 a — ^. Find its area. 11. Find the distance between the points of intersection of the curves 3 a: - 2 y + 12 = 0, and a;2 + ?/2 = 9. 12. Does the locus of ?/2 = 4 x intersect the locus of 2 a: + 3 ?/ + 2 = 0? 13. Does the locus of x"^ — ^y-\-^: = ^ cut the locus of a:2 + 2/2 = l ? 14. For what values of m will the curves a:2 + ?/2 iz: 9 and ?/ = 6 a: + m not intersect? (cf. Art. 9.) Trace these curves. 15. For what value of h will the curves ?/2 = 4x and y — x -\-^ inter- sect in two distinct points? in two coincident points? in two imaginary points (i.e., not intersect)? 16. Find those two values of c for which the points of intersection of the curves ?/ = 2 a: + c and x"^ -\- y^ — 25 are coincident. 17. Find the equation of a curve which passes through all the points of intersection of xP- + 1/2 — 25 and y'^ = ^x. Test the correctness of the result by finding the coordinates of the points of intersection and sub- stituting them in the equation just found. 60 ANALYTIC GEOMETRY [Ch. Ill 41. 18. "Write an equation which shall represent the combined loci of (1), (2), and (3) of Art. 37. Discuss and construct the loci of the equations : 19. (a-2 - y-^) {y - tan x) = 0. 22. y = x\ 25. p = a^ cos 2 0. 20. x^ - y^ = 0. 23. y^ = x^. 26. p = 3 0. 21. a;4 - y^ = 0. 24. y = 10^ 27. p = a sin 2 ^. 28. Show that the following pairs of curves intersect each other in two coincident points ; i.e., are tangent to each other. 3/2- lOz- 6y-31 =0, (a) . (9 9^2 _ 4^2 + 54:3;- IQy + 29 = 0, 2 ?/ — 3 a; + 5 = 0. 29. Find the points of intersection of the curves ^ + f.= 1 and ^-y-=l. 25 9 25 9 CHAPTER IV THE EQUATION OF A LOCUS 42. The equation of a locus. The second fundamental problem of analytic geometry is the reverse of the first (cf. Art. 31), and is usually more difficult. It is to find, for a given geometric figure, or locus, the corresponding equation, i.e., the equation which shall be satisfied by the coordinates of every point of the given locus, and which shall not be satisfied by the coordinates of any other point. The geometric figure may be given in two ways, viz. : (1) As a figure with certain known properties ; and (2) As the path of a point which moves under known conditions. In the latter case the path is usually unknown, and the complete problem is, first to find the equation of the path, and then from this equation to find the properties of the curve. This last is the third problem mentioned in Art. 31. The two ways by which a locus may be " given " corre- spond to the two conceptions of a locus mentioned in Art. 35, and they lead to somewhat different methods of obtaining the equation. The first method may be exemplified clearly, and most simply, by first considering the familiar cases of the straight line and the circle. 43. Equation of straight line through two given points.* Let Pj = (3, 2), and P^ = (12, 5) be two given points ; and * See also Art. 51. 61 62 ANALYTIC GEOMETRY [Ch. IV. let P = (a:, y) be ani/ other point on the line through P^ and Pg. Draw the ordinates M^P-^^ MP, and M^P^, and through Pj draw PjiV parallel to the a^-axis, meeting MP in R and ilfgPa in i^2- T J^ R 1 — --- 1 1 -p- If M, Fig. 28.^ M, ■n The triangles P^RP and P^R^P^^ are similar, hence RP__P^R i^2^2 Pji^g' ^■^" iff2P2 - i^/jPi Oifg - OM^ Substituting for MP, OM, M^P^, OM^, etc., their values, this equation becomes ?/- 2 rr- 3 which reduces to 5 _ 2 12-3' 3^ _ir- 3 = 0. (1) This is the required equation of the straight line through Pj and Pg, because it fulfills both the requirements of the definition [cf. Art. 35 (1)]; ^•e., it is satisfied by the coordi- nates of any (i.e., of every) point of this line, because x, y are the coordinates of any such point ; and it i% not satisfied by the coordinates of any point which is not on this line, because the corresponding constructions for such a point would not give similar triangles, and hence the proportions which led to this equation would not be true. That equation (1) is not satisfied by the coordinates of 43-44.] THE EQUATION OF A LOCUS 63 any point not on the line through Pj_ and Pg ^^7 ^^so be seen as follows : let ^3 = (2:3,^3) be any point not on the 4 is on the line PiP^i its line through P^ and P^^ the ordinate M^P^ will meet P1P2 ^^ some point P4 = (x^, y^, for which x^ = x^ but ?/4 =7^ y^. Since P coordinates satisfy equation (1), therefore 3 ^4 - ^4 - 3 = 0, .'. 3^3-2:3-3=5^0;* [since x^ = x^ and y^ ^ y^ hence the coordinates of Pg do not satisfy the equation 3^ — a? = 3. 44. Equation of straight line passing through given point and in given direction.! Let P^ = (5, 4) be the given point, let the given line through Pj make an angle of 30° with the a;-axis, and let P = (2:, y') be any other point on this line. Draw the ordinates M^P^ and ifeTP, and, through P^, draw P^B, parallel to the a:-axis to meet MP in R. Then MP - M^P^ tan RP^P = p p OM - OM^ * This proof shows clearly that if the coordinates of any point on the straight line through Pi and P^ are substituted for x and y in equation (1) the first member will be equal to zero ; if the coordinates of any point below this line are so substituted the first member will be negative ; and if the coor- dinates of any point above this line are so substituted the first jn ember will be positive. This line may then be regarded as the boundary which separates that part of the plane for which 3?/ — x — 3 is negative from the part for which this function is positive. Because of this fact that side of this line on which P3 lies may be called the negative side, and the other the positive side. t See also Art. 53. 64 ANALYTIC GEOMETRY [Ch. IV. Substituting for M^P^, MP, OM^, OM, and angle RP^P their values, and remembering that tan 30° = — - = ^ V8, this equation becomes -U-A. V i.e., x-VS^-5 + 4V3 = 0/ Fig. 30.— The equation just found is satisfied by the coordinates of any point on the given line, but is not satisfied by the coor- dinates of any point that is i^ot on this line (cf. Art. 43); hence it is the equation of the line (cf. Art. 35). 45. Equation of a circle ; polar coordinates, f In deriving this equation, let polar coordinates be employed, merely for variety, and let the pole be taken on the circumference, with a di- ameter OA extended for the ini- tial line. Let P ^ (p, ^) be any point on the circle,^ and let r be the radius of the circle. j^iG^si^ Connect P and A by a straight * The positive side of this line is tliat side on which the origin lies (cf. foot-note, Art. 43). t See also Art. 98. {Except in plane geometry, the word "circle" is employed by most writers on mathematics to mean "circumference of a circle." It will be so used in this book. 44-46.] THE EQUATION OF A LOCUS 65 line ; then, iu triangle A OP, angle OP A is a right angle, A 0P = e, OP = p, and OP:OA = cos ; i.e., p : 2r = cos ; hence p =2r cos 0. . . . (1) Equation (1) is satisfied by the polar coordinates of every point on the circle ; but is not satisfied by the coordinates of a point Q not on the circle, since angle AQO is not a right angle. Therefore Eq. (1) is the equation of this circle (cf. Art. 35). EXERCISES 1. Find the equation of the straight line through the two points (1, 7) and (6, 11) ; through the points (~2, 5) and (3, 8). Which is its posi- tive side of these lines ? 2. Find the equation of the straight line through the two points (2, 3) and (-2, -3). Through what other point does this line pass? Does the equation show this fact ? 3. Find the equation of the straight line through the point (5, ~7), and making an angle of 45° with the a:-axis ; making the angle —45° with the a:-axis. 4. Find the equation of the line through the point (~6, -2), and making the angle 120° with the a:-axis. 5. Construct the circle whose equation is p = 10 cos 0. 6. With rectangular coordinates, find the equation of the circle of radius 5, which passes through the origin, and has its center on the X-axis. Is its positive side outside or inside ? 46. Equation of locus traced by a moving point. In the problems given above, the geometric figure in each case was completely known ; and, in obtaining its equation, use was made of the known properties of similar triangles, triangles inscribed in a semicircle, and trigonometric functions. In only a few cases, however, is the curve so completely known ; in a large class of important problems, the curve TAN. AN. GEOM. — 5 66 ANALYTIC GEOMETRY [Ch. IV. is known merely as the path traced by a point which moves under given conditions or laws. Such a curve, for instance, is the path of a cannon ball, or other projectile, moving under the influence of a known initial force and the force of gravity. Another such curve is that in which iron filings arrange themselves when acted upon by known magnetic forces. The orbits of the planets and other astronomical bodies, acting under the influence of certain centers of force, are important examples of this class of "given loci." In such problems as these, the method used in Arts. 43 to 45, cannot, in general, be applied. A method that can often be employed, after the construction of an appropriate figure, is: (1) From the figure, express the known law, under which the point moves, by means of an equation involving geo- metric magnitudes ; this equation may be called the " geo- metric equation." (2) Replace each geometric magnitude by its equivalent algebraic value, expressed in terms of the coordinates of the moving point and given constants ; then simplify this algebraic equation, and the result is the desired equation of the locus. 47. Equation of a circle : second method. To illustrate this second method of finding the equation of a locus, con- sider the cirx^le as the path traced by a point which moves so that it is always at a given constant distance from a fixed O point. From this definition, find its ""^P ^ equation. Let C = (2>. 2) be the sfiven fixed point, and let P =(x^ ?/) be a point that moves so as to be always at the distance 21 from C. Then CP = j, . . . [geometric equation] 46-48.] THE EQUATION OF A LOCUS 67 but CP = V{x-Sy-{-(y -2)2 (Art. 26, [2]), . • . V(a: — 8)2 4- (^ _ 2)2 = A; [algebraic equation] ^.^., (a; -3)2+ (y- 2)2=2/- ; hence ^x^ + 4:i/^ - 24:x -I67/ -j- 21 = 0, which is the required equation. The locus of this equation can now be plotted by the methods of Art. 37, and its form and limitations can be discussed as is there done for other equations. EXERCISES 1. Find the equation of the path traced by a point which moves so that it is always at the distance 4 from the point (5, 0). Trace the locus. 2. Find the equation of the path traced by a point which moves so that it is always equidistant from the points ("2, 3) and (7, 5) (cf. Ex. 9, p. 34). 3. A line is 3 units long; one end is at the point (-2, 3). Find the locus of the other end (cf. Ex. 8, p. 34). 4. A point moves so as to be always equidistant from the y-axis and from the point (4, 0). Find the equation of its path, and then trace and discuss the locus from its equation. 5. A point moves so that the sum of its distances from the two points (0, Vo), (0, -VS) is always equal to 6. Find the equation of the locus traced by this moving point. 6. A point moves so that the difference of its distances from the two points (0, VH), (0, ~V5) is always equal to 2. Find the equation of the locus traced by this moving point. 48. The conic sections. Of the innumerable loci which may be given by means of the law governing the motion of the generating or tracing point, there is one class of par- ticular importance ; and it is to the study of this important class that the following pages will be chiefly devoted. These curves are traced hy a point which moves so that its distance 68 ANALYTIC GEOMETRY [Ch. IV. from a fixed point always hears a constant ratio to its distance from a fixed straight line. These curves are called the Conic Sections, or more briefly Conies, because they can be obtained as the curves of intersection of planes and right circular cones ; * in fact, it was in this way that they first became known. The last three examples just given belong to this class, although it is only in No. 4 that this fact is directly stated. These loci are the parabola, the ellipse, and the hyperbola ; it will be shown later that they include as spe- cial cases the straight line and the circle, f They are of primary importance in astronomy, where it is found that the orbit of a heavenly body is a curve of this kind. The general equation, which includes all of these curves, will now be derived, and the locus briefly discussed ; in a subsequent chapter will be given a detailed study of the properties of these curves in their several special forms. (a) The equation of the locus. Let F be the fixed point, — the focus of the curve ; B'D the fixed line, — the directrix of the curve ; and e the given ratio, — the eccentricity of the curve. The coordinate axes may of course be chosen as is most convenient. Let D'D be the ?/-axis, and the perpendicu- lar to it through F, i.e., the line OFX, be the cr-axis. Let P = (a;, ?/) be any position of the generat- ing point, and let OF, the fixed distance of the focus from the directrix, be denoted by k ; then the coordinates of the focus are (k, 0). Connect F and P, and through F draw LP perpendicular to the directrix. Then FP : LP = e, [geometric equation] Y D L --»¥* T^ / / X F 1 D Fig. 33. * See Note D, Appendix. t See Note C, Appendix. 48.] THE EQUATION OF A LOCUS 69 but FP = ^{x - kf + / (Art. 26), and LP = x ; [algebraic equivalents] hence V(a; — k)"^ + y'^ = ex; i.e., (l-e^)3^-\-y^-2kx + k^ = 0, . . . (1) whicli is the equation of the given locus. This equation is of the second degree ; in a later chapter it will be shown that every equation of the second degree between two variables represents a conic section. On this account it is often spoken of as the "second degree curve." (6) Discussion of equation (1). If a; = 0, then y=±k V— 1, which shows that this curve does not intersect the ?/-axis as here chosen; ^.e., a conic does not intersect its directrix. If 3/ = 0, then (1 - e2)^2 _ 2 kx -{- k"^ = 0, whence x = , or a; = :; , . . . (2) 1 -\- e 1 — e i.e., a conic meets the line drawn through the focus and per- pendicular to the directrix (the 2:-axis as here chosen) in k two points whose distances from the directrix are and k . . . 1+^ I _ respectively ; these points are called the vertices of the conic. Equation (1) shows that for every value of x, the two corresponding values of y are numerically equal but of opposite signs, hence the conic is symmetrical with regard to the a;-axis as here chosen. For this reason the line drawn through the focus of a conic and perpendicular to the directrix is called the principal axis of the conic. The form of the locus of equation (1) depends upon the value of the eccentricity (e) ; if e = 1, the conic is called a 70 ANALYTIC GEOMETRY [Ch. IV. parabola ; if e < 1, an ellipse ; and if e > 1, an hyperbola. Each of these cases will now be separately considered. (1) The parabola^ Fig. 34. e = l. If e = l, then FF:LF = 1, ^.e., FP = LP for every position of the tracing point,* hence the curve passes through J., — the point mid- way between and F^ — but does not again cross the principal axis (cf. also equations (2), above). Moreover, when e = 1, equation (1) becomes I.e.. / - 2 ^rc + ^2 ^ 0, / = 2 klx — -J, (3) which is the equation of the parabola, the coordinate axes being the principal axis of the curve and the directrix. Equation (3) shows that there is no point of this parabola for which a^<-, and also that y changes from to ± go when X increases from - to oo ; hence the parabola recedes indefinitely from both axes in the first and fourth quadrants. Its form is given in Fig. 34. (2) The ellipse^ e <1. Equation (1) may be written in the form k \f h ^2= (1-^2) — X X 1 + e (4)t * This property enables one to construct any number of points lying on the parabola, thus : with F as center, and any radius not less than \0F^ describe a circle, then draw a line parallel to OY and at a distance from it equal to the chosen radius ; the points in which this line cuts the circle are points on the parabola. Other points can be located in the same way. See also Note B, Appendix. t Equation (4) enables one to construct any number of points on the 48.] THE EQUATION OF A LOCUS 71 which shows, e being less than 1, that ?/ is imaginary for all values of x except those which satisfy the condition k _ - Jc X l^e< <\-e' hence the ellipse lies wholly on the positive side of its direc- trix, and between two lines which are parallel to the directrix Y p and distant from it and l-\-e 1-e tion (4) shows that as x increases from respectively. Equa- k . k 1 + e to ^ are ellipse. E.g. ,\etx= OM ; then the factors (x ] and ( x] \ 1 + e/ \l-e J the two segments AM and MA' of the line AA'^ and geometri- cally their product equals the square of the ordinate 31Q of the semicircle of which A A' is the diameter. If now the point P on MQ be so constructed that MP = Vl - e2 • MQ, then P is a point on the ellipse whose equation is (4) above. Similarly, any number of points on the curve can be constructed. This method shows also that the ordinates of an ellipse are less than, but in a con- stant ratio to, the corresponding ordinates of the circle of which the diameter is the line joining the vertices of the ellipse. See also Note B, Appendix. 72 increases from to k ANALYTIC GEOMETRY ek [Ch. IV. Vl - e2 V [which value it reaches when X = J and then decreases again to 0. The form of the curve is therefore as shown in Fig. 35, where 0F= k, OA = J^, 00 = ^-^. OA = J^, and CB =^ ^^ 1+e 1-e^ 1-e Vl-e=i (3) The hyperhola^ e>l. Equation (1) may also be written in the form ^2 — ^g2 _ l^(x — k 1 + e X — k 1-e (5) which, when e>l. shows that ?/ is imaginary for all values of X between x = and x = , and that y is real for 1+e 1-e ^ all other values of x. Equation (5) also shows that, as x increases from k to GO, ?/ changes from to ±co, and k 1 + e that, as x decreases from 1 — e The form of the curve is therefore as show^n in to — GO, ?/ changes from to ± 00. k k k Fig. 37, where OA = -^^^— and OA' = ^ 1 + e 1-e e-1 Although these three curves differ so widely in form, they are really very closely related as will be further shown in Chap. XII, and in Note D of the Appendix. 48-49.] THE EQUATION OF A LOCUS 73 49. The use of curves in applied mathematics.* In Chap- ter III it was shown that whenever the relation between two variables, whose values depend upon each other, can be defi- nitely stated, ^.e., when the variables can be connected by an equation, then the geometric or graphic representation of this relation is given by means of a curve. Such a curve often gives at a glance, information which would otherwise require considerable computation to secure ; and in many cases it brings out facts of peculiar interest and importance which might otherwise escape notice. The use of graphic methods in the study of physics and engineering, as well as in statistics and many other branches of investigation, is already extensive and is rapidly increas- ing. Under the name " graphic methods " there are in- cluded, however, not only such examples as those already given, where the equation connecting the variables is known, but also those where no such equation can be found ; in these latter cases the curves constitute almost the only prac- tical way of studying the relations involved. As a simple example of this kind, suppose the temperature, of a patient to be accurately observed at intervals of one hour ; if the numbers representing the hours, i.e.^ 1, 2, 3, ••• are taken as abscissas, and the corresponding numerical values of the temperatures be taken as ordinates, then a smooth curve drawn through the points so determined will express graphi- cally the variation of the temperature of this patient with the time. This curve will also show to the physician what was the greatest and least temperature during the inter- val of the observations, as well as the time when each of * For most of the suggestions in this article, and in the examples that follow it, the authors are indebted to Mr. J. S. Shearer of the Department of Physics of Cornell University 74 ANALYTIC GEOMETRY [Ch. IV. these was attained. In this problem the curve gives no new information, but it presents in a much more concise and forcible form the information given by the tabulated numbers. Again, if the distances passed over by a train in successive minutes during the run between two stations are taken as ordinates, and the corresponding number of minutes since starting, as abscissas, a smooth curve drawn through the points so determined will show at a glance, to an experi- enced eye, where and when additional steam was turned into the cylinders, brakes applied, heavy grades encountered, etc., etc. In all such cases the coordinates of the points are taken to represent the numerical values of related quantities, such as time, length, weight, velocity, current, temperature, etc., and the curve through the points so determined usually gives, to an experienced person, all the information concerning the relations involved that is of practical importance. It is in the study of such curves that much of the value of train- ing in analytic geometry becomes apparent to the physicist and the engineer. The student should early learn to trans- late physical laws into graphic forms, and he should give careful attention to the interpretation of all changes of form, intercepts, intersections, etc., of such curves. EXERCISES 1. In simple interest if jo = principal, t=time, ?' = rate, and a = amonnt, then a =p (1 + rt). K now particular numerical values are given to p and r, and if the values of the variable a be taken as ordinates, and the corresponding values of t as abscissas, then the locus of this equa- tion may be drawn. Draw this locus. What line in the figure repre- sents the principal? What feature of the curve depends upon the rate per cent ? Interpret the intercepts on the axes. 49.] THE EQUATION OF A LOCUS 75 2. Give to p and r in exercise 1 different values and, with the same axes, draw the corresponding locus. How do these loci differ? What does their point of intersection mean ? 3. With the same axes as before draw the curve for which interest and time are the coordinates; how is it related to the curves of exercises 1 and 2? 4. Draw and discuss the curve showing the relation between amount^ principal, rate, and time in the case of compound interest. (a) When interest is compounded annually. (^) When interest is compounded quarterly, (y) When interest is comx^ounded instantaneously. 5. A wage earner has akeady been working 10 days at |1.50 per day, and continues to do so 20 days longer, after which he is idle during 8 days ; he then works 14 days more at the same wages, after which his employer raises his wages to .*$2.50 per day for the next 20 days : using the amounts earned as ordinates, and the time (in days) as abscissas, draw carefully the broken line which states the above facts. What modification of the drawing would be necessary to show that the wage earner forfeited 50 cents per day during his idleness ? 6. The following table shows the production of steel in Great Britain and the United States from 1878 to 1891.* U.S. G.B. U.S. G.B. 1878 . . 7.3 (100,000 long tons) 10.6 (100,000 long tons) 1885 . . 17.1 19.7 1879 . . 9.3 10.9 1886 . . 25.6 23.4 1880 . . 12.5 13.7 1887 . . 33.4 31.5 1881 . . 15.9 18.6 1888 . . 29.0 34.0 1882 . . 17.4 21.9 1889 . . 33.8 36.7 1883 . . 16.7 20.9 1890 . . 42.8 36.8 1881 . . 15.5 18.5 1891 . . 39.0 32.5 Using time (in years) as abscissas, and quantity of steel produced (100,000 tons per unit) as ordinates, the separate points represented by * Taken by permission from Lambert's Analytic Geometry. 76 ANALYTIC GEOMETRY [Ch. IV. the table have been plotted (Fig. 38) and then joined by straight lines, dotted for Great Britain and full for the United States.* Interpret fully the figure. 45 40 35 30 25 20 15 10 5 / \ .y - / \ // ^ / N f ,y ^ /y / --^ ' r^ ^ /^ 1878 79 '80 "81 '83 '83 '84 '85 Fig. 38. '87 '88 '90 '91 7. Exhibit graphically the mformation contained in the following table on the expense of moving freight per "ton-mile" on N. Y. C. & H. R. R. R. from 1866 to 1893. 1866 1867 1868 1869 1870 1871 1872 2.16^ 1.95 1.80 1.40 1.15 1.01 1.13 1873 1874 1875 1876 1877 1878 1879 1.03^ .98 .90 .71 .70 .60 .55 1880 1881 1882 1883 1884 1885 1886 .54^ .56 .60 .68 .62 .54 .53 1887 1888 1889 1890 1891 1892 1893 .56;* .59 .57 .54 .57 .54 .54 8. The following table gives the population of the countries named between 1810 and 1896 : f * In the figure the linear unit on the 2C-axis is 5 times as long as the linear unit on the ?/-axis. It will, however, be noticed that the essential feature of a system of coordinates, the "one-to-one correspondence" of the symbol (x, y) and the points of a plane, is not disturbed by using different scales for ordinates and abscissas. t The authors are indebted to Professor W. F. Willcox of Cornell Univer- sity for these data, which are compiled from the Statesman'' s Year Book for 1897, and from Statistik des Deutschen Beichs, Bd. 44, 1892. 49.] THE EQUATION OF A LOCUS 77 British Isles Lands now included in the German Empire Year Population Year Population 1801 15,896,000 1816 24,831,000 1811 17,908,000 1837 31,540,000 1821 20,894,000 1847 34,753,000 1831 24,029,000 1856 36,130,000 1841 26,709,000 1865 39,399,000 1851 27,369,000 1872 41,028,000 1861 28,927,000 1876 42,775,000 1871 31,485,000 1885 46,856,000 1881 34,885,000 1895 52,280,000 1891 37,733,000 France Ireland United States Year Population Year Population Year Population 1821 30,462,000 1811 5,938,000 1810 7,240,000 1841 34,230,000 1821 6,802,000 1820 9,634,000 1861 37,386,000 1831 7,767,000 1830 12,866,000 1866 38,067,000 1841 8,175,000 1840 17,069,000 1872 36,103,000 1851 6,552,000 1850 23,192,000 1876 36,906,000 1861 5,799,000 1860 31,443,000 1881 37,672,000 1871 5,412,000 1870 38,558,000 1886 38,219,000 1881 5,175,000 1880 50,156,000 1891 38,343,000 1891 4,705,000 1890 62,622,000 1896 38,518,000 Employing the number of years as abscissas, and the population (500,000 per unit, — numbers at left of figure represent millions) as ordi- nates, the separate points represented by the above table have been plotted (Fig. 39) and then joined by straight lines. The figure gives all the information contained in the tabulated results, besides showing at a glance the relative population of the different countries at any given time. The student may account historically for the abrupt fall in the line representing the population of France; and for the gradual down- ward tendency in the line representing the population of Ireland. 78 ANALYTIC GEOMETRY [Ch. IV. 49.] THE EQUATION OF A LOCUS 79 EXAMPLES ON CHAPTER IV 1. Find the equations of the sides of the triangle whose vertices are the points (2, 3), (4, -5), (3, -6) (cf. Art. 43). Test the resulting equations by substitution of the given coordinates. 2. Find the equations of the sides of the square whose vertices are (0, -1), (2, 1), (0, 3), (-2, 1). Compare the equations of the parallel sides ; of perpendicular sides. 3. Find the coordinates of the center of the square in Ex. 2. Then find the radius of the circumscribed circle, and (Art. 47) the equation of that circle. Test the result by finding the coordinates of the points of intersection of one of the. sides with circle (Art. 39). 4. Find the equation of the path traced by a point which is always equidistant from the points (a) (2, 0) and (0, -2) ; (jS) (3, 2) and (6, 6) ; (y) (<^ + &, « — &) and {a — b, a + b). 5. A point moves so that its ordinate always exceeds f of its abscissa by 6. Find the equation of its locus, and trace the curve. 6. A point moves so thut the square of its ordinate is always 4 times its abscissa. Find the equation of its locus and trace the curve. 7. Find the equation of the locus of a point which moves so that the sum of its distances from the points (1, 3) and (4, 2) is always 5. Trace and discuss the curve. 8. Find the equation of the locus of the point in example 7, if the difference of its distances from the fixed points is always 2. 9. Express by a single equation the fact that a point moves so that its distance from the a:-axis is always numerically 3 times its distance from the ?/-axis. 10. A point moves so that the square of its distance from the point (a, 0) is 4 times its ordinate. Find the equation of its locus, and trace the curve. 11. A point moves so that its distance from the x-axis is J of its dis- tance from the origin. Find the equation of its locus, and trace the curve. 12. A point moves so that the difference of the squares of its dis- tances from the points (1, 3) and (4, 2) is 5. Find the equation of its locus and trace the curve. 80 ANALYTIC GEOMETRY [Ch. IV. 49. 13. Solve example 12 if the word "sum" is substituted for "differ- ence." 14. Let A = (a, 0), B = (b, 0), and A' = {-a, 0) be three fixed points; find the equation of the locus of the point F = (x, y) which moves so that PB' + PA^ = 2 PA'\ 15. A point moves so that ^ of its abscissa exceeds ^ of its ordinate by 1. Find the equation of its locus and trace the curve. 16. Find the equation of the locus of a point that is always equi- distant from the points (~3, 4) and (5, 3); from the points (~3, 4) and (2, 0). By means of these two equations find the coordinates of the point that is equidistant from the three given points. 17. Let A = (-l, 3), B=(-S, -3), C = (l, 2), Z) = (2, 2) be four fixed points, and let P=(x, y) be a point that moves subject to the con- dition that the triangles PAB and PCD are always equal in area; find the equation of the locus of P. 18. If the area of a triangle is 25 and two of its vertices are (5, - 6) and ("3, 4), find the equation of the locus of the third vertex. 19. A point moves so. that its distance from the pole is numerically equal to the tangent of the angle which the straight line joining it to the origin makes with the initial line. Find the polar equation of its locus and plot the figure. CHAPTER V THE STRAIGHT LINE. EQUATION OF FIRST DEGREE Ax + By + C = 50. In Chapter III it was shown that to every equation between two variables there corresponds a definite geometric locus, and in Chapter IV it was shown that if the geometric locus be given, its equation may be found. -It still remains to exhibit in greater detail some of the more elementary loci and their equations, and to apply analytic methods to the study of the properties of these curves. Since the straight line is a simple locus, and one whose properties are already well understood by the student, its equation will be ex- amined first. In studying the straight line, as well as the circle and other second degree curves, to be taken up in later chapters, it will be found best first to obtain the simplest equation which represents the locus, and to study the properties of the curve from that simple or standard equation. Then it remains to find methods for reducing to this standard form any other equation that represents the same locus. 51. Equation of straight line through two given points. A numerical example of the equation of tlie line through two fixed points has already been given in Art. 43 ; in the pres- ent article the equation of a straight line tlirough an^ two given points will be derived ; the method, however, will be precisely the same as that already employed in the numerical example. TAN. AN. GEOM. — 6 81 82 ANALYTIC GEOMETRY [Ch. V. Let the two given fixed points be Fi=(^xi, y{) and P,^ (x2, y-^^ and let P=(rr, ?/) be any other point on the line through Pi and P^. Draw the ordinates MiP^^ -^2-^*21 ^^^ M Fig. 40.— rrQ.40.^ MP \ also through P^ draw P1R2 parallel to the a;-axis, and meeting MP in R and M2P2 in R2. Then the triangles PiRP and P1R2P2 are similar ; RP__P^R . MP-M,P, _ OM- OM ^ R2P2~ PiR2 ''"" M2P2-M,P,~ OM2-O3I,' Substituting in this last equation the coordinates of Pi, P2, and P, it becomes y -y\ ^ a? - a?! ^2 - 2/1 a?2 - a?! ' • • * "-J and since P = (x^ y) is a^i?/ point on the line through Pi and P2, therefore equation [9] is satisfied by the coordinates of every point on this line. That equation [9] is not satisfied by the coordinates of any point except such as are on the line P1P2 may be proved as was done in Art. 43. Equation [9] then fulfills both requirements of the defi- nition in (1) of Art. 35, and is therefore the equation of the straight line through the two points (a^i, y{) and (2^2? ^2) • This equation will be frequently needed and will be referred to as a standard form ; it should be committed to memory.* * Throughout this book the more important formulas are printed in bold- faced type ; they should be committed to memory by the learner. 51-52.] THE STRAIGHT LINE 83 52. Equation of straight line in terms of the intercepts which it makes on the coordinate axes. If the two given ^J Y B "V. \^ ^^' ^^ \^ X Fig. 43. 4^ ^M points in Art. 51 are those in which the line cuts the axes of coordinates, i.e., A=(a, 0) and B = (0, 6) (Fig. 41), then equation [9] becomes y — _x b-0~0 a a that is, ^ + y a h 1, [10] where a and h are the intercepts which the line cuts from the axes. This is another standard form of the equation of the straight line ; it is known as the symmetrical or the inter- cept form. Equation [10] may also be derived independently of equa- tion [9] thus : let the line MN (Fig. 42), whose equation is to be found, cut the axes at the points A = (^a, 0) and ^ = (0, 5), and let P = (x,y) be any other point on this line. Connect and P; then area OPB -h area OAP = area OAB ; that is, ^hx ■\- \ ay = \ ah., X V and, dividing by \ ab, this equation becomes - + ^ = 1^ as above. EXERCISES 1. Show that equation [10] is not satisfied by the coordinates of any point except those lying on MN. 84 ANALYTIC GEOMETRY [Ch. V. 2. Write down the equations of the lines through the following pairs of points : (a) (3, 4) and (5, 2); (y) ("6, 1) and (-2, -5) ; (/?) (3, 4) and (5, "2); (8) (-15,-3) and (|, ^)- 3. Write the equations of the lines which make the following inter- cepts on the X' and y-axes respectively. (a) 4 and 7; i/3) "3 and 5; (y) | and -^; (8) -^ and 3 a. 4. What do equations [9] and [10] become if one of the given points is the origin? 5. By drawing, in Fig. 42, a perpendicular^^il/ from P to the x-axis, derive equation [10] from the similar trisLngl^MpfiP and OAB. 6. Is equation [10] true if P is on MN but |/ot between A and B'^ 7. Are equations [9] and [10] true if the coordinate axes are not at right angles to each other? 8. Is the point (3, 4^) on the line through the points (2, 3) and (5, 7) ? On which side of this line is it ? AVhich is the negative side of this line? ^ 9. What intercepts does the line through the points (1, -6) and (-3, 5) make on the axes ? 10. The vertices of a triangle are : (4, -5), (2, 3), and (3, -6). Find the equations of the sides; also of the three medians; then find the coordinates of the point of intersection of two of these medians, and show that these coordinates satisfy the equation of the other median. What proposition of plane geometry is thus proved? 11. Find the tangent of the angle (the " slope," cf . Art. 27) which the line in exercise 9 makes with the a;-axis.~ 12. Draw the line whose equation is - + ^ = 1, and then find the .J o equations of the two lines which pass through the origin and trisect that portion of this line which lies in the first quadrant. 63. Equation of straight line through a given point and in a given direction (cf. Art. 44). Let P^ = (ix^, y^) be the given point, and let the direction of the line be given by the angle XAP = 6 which the line makes with the a; -axis; also let P={x, y) be any point on the given line and denote the slope, i.e., tan ^, bj m. Draw the ordinates 52-53. J THE STRAIGHT LINE 85 M^P^ and MP, and through Pj draw P^R parallel to the a;-axis and meeting the ordinate MP in R. Then, in triangle RP^P, the angle RP^P = d ; Fig, 43. hence m tan 6 = RP P^R [11] X — x^ [Since RP = y — y^ and P^R = x — x^~\ ; that is, y -yi = in{oc — xi)f which is the desired equation. Cor. If the given point be ^=(0, 5), ^.e., the point in which the line meets the y-axis, then equation [11] becomes y = Qnx + b. . . . [12] Equation [12] is usually spoken of as the slope form of the equation of the straight line. EXERCISES 1. What do the constants m and b in equation [12] mean? Draw the line for which wi = 4 and 6 = 3; also that for which m = — 1 and 7> — _ 3 2. What is the effect on the line [12] of a change in b while m remains the same? What if 771 be changed and b left unchanged? 3. Describe the effect on the line [11] of changing ??i while x-^ and y^ remain the same ; also the effect resulting from a change in x^ while m and ?/^ remain the same. 4. Write the equation of a line through the point (~3, 7), and mak- ing with the X-axis an angle of 30'^; of -30°; of (=:^y'^ of /^Zj"^^'"'* 5. Write the equations of the following lines : (a) slope 3, y-intercept 8 ; (/?) slope i, 3/-iutercept "3 ; (y) slope -2, ^/-intercept ~|. 86 ANALYTIC GEOMETRY [Ch. V. 6. A line has the slope 6 ; what is its y-intercept if it passes through the point (7, 1) ? 7. What must be the slope of a line whose ?/-intercept is ~3, in order that it may pass through the point (~5, 5) ? 8. Is the point (1,2) 01^ the line passing through the point (~2, ~14), and making an angle tan~i^ with the a:-axis? 9. How do the lines y = ^x — 1, y = ^x-\-'l, and 2?/ — 6a: + 15 = differ from each other? AVhat have they in common? Draw these lines. 10. What is common to the lines 3/ = 3 a; — *1, 2 3/ = 5 a; — 2, and 7a;-32/ = 3? 11. What is the slope of [9] ? of [10] ? 12. Derive equation [12] independently of equation [11]. 54. Equation of straight line in terms of the perpendicular from the origin upon it, and the angle which that perpendicular makes with the i^-axis. Let HKhe the line whose equation K Y ^^ a is sought, and let the perpendicular (^OIi= p) from upon this line, and the angle (a) which this perpendicular makes with the a;-axis, be given. Also let P = (a:, z/) be any point on ffK; then by projection upon ON (Art. 17), OM cos a -\- MP sin a = ON, ^^e., xcosa + ysma= p, . . . [13] which is the required equation. Equation [13] is known as the normal form of the equa- tion of the straight line. In the following pages p will always be regarded as posi- tive, and a as positive and less than 860°. 53-55.] THE STRAIGHT LINE 87 55. Normal form of equation of straight line : second method. The student should bear in mind tliat to get the equation of a curve, lie has merely to obtain an equation that is satisfied by the coordinates of every point on the curve, and not satisfied by tlie coordinates of any other point ; and that it is wholly immaterial what particular geometric property he may employ in the accomplishment of this purpose. This fact is already illustrated in Art. 52, where equation [10] was obtained in two ways, while Ex. 5, p. 84, gives still a third method by which the same equation may be found. So also it is possible to derive equation [13] by other methods than that employed in Art. 54.* U.g., in Fig. 41 draw a perpendicular from to the line AB, let its length be denoted by p, and let a be the angle which it makes with the a:-axis, then a cos a = p, and b sin a= p, whence a = ^ , and b = . . cos a sin a Substituting these values of a and b in equation [10], it becomes ^ y H — ^— = 1, i.e., X cos a -{- y sin a= p, p p cos a sin a which is the form already derived in Art. 54. Note. In Art. 2, constants, variables, etc., were illustrated by means of a triangle. Now that the student has learned that the equation - + ^ = 1, for example, represents a straight line, i.e., that this equation a b is satisfied by all those pairs of values of x and y which are the coordi- nates of points on this line, a somewhat better illustration can be given. Both X and y are variables, but are not independent ; each is an implicit function of the other. For any particular line a and h are constants, but they may represent other constants in the equation of another line, i.e., they are arbitrary constants, and are often called parameters of the line. * See also Ex. 6 below. 88 ANALYTIC GEOMETRY [Ch. V. EXERCISES 1. The perpendicular from the origin upon a certain line is 5 ; this perpendicular makes an angle of - with the a:-axis ; what is the equation of the line ? 2. If in equation [13] p is increased while a remains the same, what is the effect upon the line ? If a be changed while p remains the same, what is the effect ? 3. A certain line is 3 units distant from the origin, and makes an angle of 120° with the a;-axis ; what is its equation ? 4. Given a = 30°, what must be the length of p in order that the line HK (see Fig. 44 a) shall pass through the point (7, 2) ? 5. A line passes through the point (~3, -4), and a perpendicular upon it from the origin makes an angle of 45° with the x-axis. What is the equation of this line ? 6. In Fig. 44 a draw through M a line parallel to HK, meeting ON in /?; then draw through P a perpendicular to MR, meeting it in Q; by means of the figure so constructed derive equation [13] anew. 56. Summary. The results of Arts. 51-55 may be briefly summarized thus : The position of a straight line is determined by ; (1) two points through which it passes ; (2) one point and the direc- tion in which the line passes through this point. Under (1) there is the special case in which the two given points are one on the a;-axis and the other on the y-axis. Under (2) there are two special cases : (a) when the given point is on an axis (the ?/-axis say), and (/3) when the point is given by its distance and direction from the origin, while the line whose equation is sought is perpendicular to the line which connects the given point to the origin. Corresponding to these two general and three special cases, there have been derived five standard forms of the equation of the straight line, viz.: equations [9], [10], [11], [12], and [13]. It may be remarked that equations [9] and [10] are inde- pendent of the angle between the coordinate axes, while [11], 4 55-57.] THE STRAIGHT LINE 89 [12], and [13] (m, a, and^ retaining their present meanings) are true only when the axes are rectangular. It may also be pointed out that, from the nature of its derivation, equa- tion [9] is inapplicable when the line is parallel to either axis ; equation [10] is inapplicable when the line passes through the origin ; and equations [11] and [12] are not applicable when the line is parallel to the ?/-axis. 57. Every equation of the first degree between two variables has for its locus a straight line. It will probably not have escaped the reader's notice that the five " standard " equa- tions (equations [9] to [13]) of the straight line, which have been derived in Arts. 51 to 54, are each of the first degree. It will now be shown that every equation of the first degree between two variables has a straight line for its locus. The most general equation of this kind may be written in the foi"m Ax + By-\-C=0, . . . (1) where A^ B^ and C are constants, and neither A nor B is zero.* Let Pi=(x^, ^i), P2 = (^2' ^2)' aii^ ^3 = (% ^3) be any three points on the locus of equation (1). Draw the ordi- nates M^P^, M^P^^ and M^P^\ also draw HP,^ and KP^ parallel to the a:-axis. sJ^ Then, by Art. 35 (1), /Ng A..3+%3 + (7=0...(4) L'^ I I ^3^< / Fig. 45. * If either A or B, say A^ is zero, then the equation may be written in the C form : y = , which is the equation of a straight line parallel to the x-axis, ^ C and at the distance from it [cf. Art. 33, (2)]. 90 ANALYTIC GEOMETRY [Ch. V. By subtracting eq. (3) from eq. (2), and also eq. (4) from eq. (3), the two equations and ^(^2 -^'3) + ^(^2 -^3)=^' are obtained. These give a^l2^-i and ^^^^3^-4; ... (5) hence, VlZll = yJLlll, ... (6) 2^1 — ^2 ^^2 *^3 But t/i - ^2 = ^Pv ^1 - ^2 = -^1^2 = -^^2^ y<2,-yz = KPv ^^^^ ^2 - ^3 = -^2^3 = --^A ' hence, from eq. (6), -=pi = j[p' Also, by construction, hence, triangle HP^P^ is similar to triangle KP^P^, and Z.P^P^E = /.PJP^K; . • . Z PiP2S^ + Z^P^^ + ZirP2P3 = Z P^P^K + Z. P^KP^ +Z XP2P3 = 2 rt. Zs; ^.e., P2 lies on the straight line joining P^ and P3. But, since P^ is any point on the locus of Ax-\- By + C = ^, hence all points of this locus lie on the same straight line P^P^, which, therefore, constitutes the locus of Ax -\- By + C =^. Since this demonstration does not depend upon the angle ft), therefore it applies whether the axes are oblique or rec- tangular ; hence the theorem : every equation of the first degree- between two variables, ivJien interpreted in Cartesian coordinates, represents a straight line.* * This conclusion may also be drawn thus : clear equation (6) of frac- tions, transpose all the terms to the first member, and multiply by ^ sin w ; 67-58.] THE STRAIGHT LINE 91 Because of this fact, such an equation is often spoken of as a linear equation. N'oTE. In the equation Ax -\- By + C = 0, there are apparently three constants; in reality, there are but two independent constants, viz. the ratios of the coefficients (cf. Art. 88). This corresponds to the fact that a straight line is determined geometrically by two conditions. 58. Reduction of the general equation Aoc + By +C = to the standard forms. Determination of «, b, m, i>, and a in terms of A, B, and O.* (1) Reduction to the standard form - + ^ = 1 (^symmetric or intercept form). That the equation Ax + By-\-C = . . . (1) represents some straight line lias just been shown (Art. 57) ; again, since multiplication by a constant, and transposition, do not change the locus (Art. 38), therefore "" +-^=lt . . . (2) A B represents the same line. But equation (2) is in the re- quired form (Art. 52), and its intercepts are : a = — 7, and 6=— — . A B (2) Reduction to the standard form y = mx + h Qslope form). the resulting equation asserts [see Art. 29, (1)] that the area of the triangle formed by the points Pi, Pj, and P3, is zero; i.e., these three points lie on a straight line ; but they are any three points on the locus of Ax + By-{- C =0, hence that locus is a straight line. * These reductions constitute a second proof of the theorem of Art. 57. t If O = 0, the line represented by (1) goes through the origin, and the symmetric form of the equation is inapplicable (Art. 56) ; but, in that case, the above reduction also fails, since it is not permissible to divide the n^pn- bers of an equation by zero. 92 ANALYTIC GEOMETRY [Ch. V. The equation Ax + By + (7=0 has the same locus as. has the equation (see Art. 38); but this is the equation (Art. 53) of a line drawn through the point fO, — ^), and making with the 2:-axis the angle = tan ~ M — ^ ) 5 hence equation (3) is in the required form, and m = — — , and 6 = — — . (3) Reduction to the standard form xcos a -\- y sin a = p (normal form). If equation (1) and X cos a -{- y sin a = p . . . (4) represent the same line, then they differ merely by some constant multiplier, say k (cf. Art. 38). Then kAx + kBy + kC = x cos cc + y sin a ~ p = ; . • . kA = cos a, kB = sin a, and kC = —p ; . • . k'^A^ + k^B^ = cos ^a + sin ^a = l; whence k = — " ; hence cos a = — . sm a = and p V^2^^2 -VA^ + B^ O * If 5 = 0, the line represented by equation (1) is parallel to the y-axis, ai* the slope form of the equation is inapplicable (Art. 56) ; but, in that case, the above reduction also fails. 58.] THE STRAIGHT LINE 93 wherein the algebraic sign of VjL^ + B^ is to be chosen so as to make — — z^^^^ positive, since p is to be always posi- tive (Art. 54); ^.e., the sign of VJ.^ + B^ is to be opposite to that of the number represented by C. Hence, to reduce equation (1) to the normal form, ^.e., to the form of equation (4), it is only necessary to divide equa- tion (1) by V^2 + B^, with the sign properly chosen, and transpose the constant term to the second member. This gives A , B -C V J.2 + B^ VW+W V^2 4_ ^2 (4) Another method for reduction to the normal form. If the equation Ax + By + C = and x cos a + y sin a= p represent the same line, then they must have the same ^-intercept and the same slope, i.e,^ (5) and — ^ = ; • • • C'^) B sm a Squaring eq. (6), and adding 1 to each member, gives A^ + B^ _ cos^ a + sin^ a B^ sin^ a B P sin a A cos a B sin a sma = sm-^a B A — C whence cos a = ^ and p = — . 94 ANALYTIC GEOMETRY [Ch. V. as before. These, then, are the values of p, sin a, and cos a, which are to be substituted in x cos a-\- y sin a =p. Hence ^ r. ^ . ^ -v— ^ V^2 ^ ^2 V^2 + ^2 V J12 + J52' is an equation representing the same locus as Ax + By + (7= 0, and having the normal form. 59. To trace the locus of an equation of the first degree. In Art. 57 it was proved that the locus of an equation of the first degree in two variables is a straight line ; but a straight line is fully determined by any two points on it ; hence, to trace the locus of a first degree equation it is only necessary to determine two of its points, and then to draw the indefinite straight line through them. The two points most easily determined, and plotted, are those in which the locus cuts the axes ; they are therefore the most advantageous points to employ. If the line is parallel to an axis, then only one point is needed. E.g.^ to trace the locus of the equation the ordinate of the point in which this line crosses the rr-axis is ; let its abscissa be x-^, then (x-^^ 0) must satisfy the equa- tion 2a;-32/ + 12 = 0; hence 2:^1-3.0 + 12 = 0, whence ar^ = — 6, i.e.^ the line crosses the a;-axis at the point (~6, 0). In like manner it is shown that it crosses the y-axis at the point (0, 4). Therefore LM is the locus of 2 2: - 3 ?/ + 12 = 0. 58-60.] THE STRAIGHT LINE 95 60. Special cases of the equation of the straight line Aoc + By -\- C = O. This equation, written in tlie intercept form [Art. 58 (1)] becomes X y C '^ n~ '' * * * ^ -^ ~A ~B If in equation (1), A is made smaller and smaller in com- parison with (7, then the a^-intercept [ — -] becomes larger and larger ; if ^ = in comparison with (7, the 2:-intercept grows infinitely large, the line (1) becomes' parallel to the a;-axis, and its equation becomes ^ ^ y -x . : . .. ^ — + -^ = 1; i.e., y = - B C""' ....,^---g, which agrees with the foot-note of Art. 57. Similarly, if ^ = in comparison with C^ the line (1) be- comes parallel to the ?/-axis, and its equation becomes . = --. If both A and B approach zero simultaneously in compari- son with C, then both the intercepts become indefinitely large, and the line (1) recedes farther and farther from the origin. In accordance with what has just been said, a line that is wholly at infinity might have its equation written in the form . a: 4- . ^ + (7 = 0, . . . (2) or, as it is sometimes written, (7=0; . . . (3) but equations (2) and (3) are merely abbreviations for the statement : " As both A and B approach zero in comparison with (7, the line moves farther and farther from the origin." 96 ANALYTIC GEOMETRY [Ch. Y. EXERCISES 1. Reduce the following equations to the intercept (symmetric) form, and draw the lines which they represent : (a) 3x-2z/ + 12 = 0; (^8) 3x - 2 z/ + 1 = 5a; + 3; (y) 2y = lb-y+^X', . (8) ^Jzl^jJ. = 9. ^+ i y 2. Reduce to the slope form, and then trace the loci : (a) 7a:-53/ + 6(?/- 3a:)=- 10a;4-4; (yg) 3a: + 2?/ + 6 = 0; (y) 3a:+5 = 3-2^. Which is the positive side of the line (^) ? (cf. foot-note, Art. 43.) 3. Reduce to the normal form, and then trace the loci : (a) 3a: + 4?/ = 15; (yg) 3a; - 4?/ + 15 = ; (y) a; - 3 ?/ = 5 + 6 a; ; (8) ^x = y -b. 4. Show that the lines 3 a: + 5 = ?/ and 6 a: — 2 ?/ = 81 are parallel. 5. What is the slope of the line between the two points (3, -1) and (2, 2) ? What is its distance from the origin ? Which is its negative side? 6. A line passes through the point (5, 6) and has its intercepts on the axes equal and both positive. Find its equation and its distance from the origin. 7. A straight line passes through the point (1, -2) and is such that the portion of it between the axes is bisected by that point. What is the slope of the line? 8. What are the intercepts which the line through the points (-1, 3) and (6, 7) makes on the axes ? Through the points («, 2 a) and {h, 2h)l 9. What system of lines obtained by varying the parameter h is rep- resented by the equation y = 6 a: + & ? 10. What system of lines obtained by varying the parameter m is represented by the equation y = mx + 6 ? 11. What family (system) of lines obtained by varying the parameter a is represented by the equation x cos a + y sin a = 5 ? To what curve is each line of the family tangent ? 12. Find cos a and sin a for the lines (a) y = mx + h, (/S) - + {=1, a (y) ? = ?, (8) 7a:-5^+l=0. 60-61.] THE STRAIGHT LINE 97 13. Find by means of cos a and sin a what quadrant is crossed by each of the lines : (a) 3x + 2 = 2y; (^) 5a: + 3y + 15 = 0; (y) x-y/^y-\0 = 0. 14. What must be the slope of the line '^x - ky = 11 in order that it shall pass through the point (1, 3)? Can k be determined so that the line will pass through the origin? 15. Determine the values oi A, B, C in order that the line Ax -{- By -\- C = shaU pass through the points (3, 0) and (0, -12). [Art. 57, Note.] 16. Derive equation [9] by supposing (x^, y{) and (a-'g, 2/2) ^^ ^® ^^^ points on the line y = mx + &-; and thence finding values for m and h. 17. Find the slopes of the lines 2 ?/ — 3 a; = 7 and 32/ + 2x — 11 = 0; and thence show that these lines are perpendicular to each other. 18. Find cos a for each of the lines 7a; + ?/ — 9 = and x — 7y + 2 = 0, and then show that the two lines are perpendicular to each other. ' 19. Show by means of: (1) the slopes; (2) the angles; that the lines 2y-3x=:7, 2y-3a; + 5 = 0, 10y-15x + c = are all parallel. 20. Reduce the equation Ax -{- By +C =0 to the normal form, i.e., to the form x cos a + y sin a=p. Suggestion : the two equations as representing the same line, make the same intercepts on the axes. 61. To find the angle made by one straight line with another. Let the equations of the lines be i/ = mjX-\-hi, , . . (1) and ^ = vYiiX + 52,... (2) where mi = tan By-, ^2 = tan ^2? and ^1, ^2 are the angles which these lines make, respec- tively, with the a^-axis. It is required to find the angle 0, measured from line (2) to line (1). TAN. AN. GEOM. — 7 \ X 98 ANALYTIC GEOMETRY [Ch. V. Since <^ = ^i — O2', , , tan 6x — tan 62 ^ a j. -1 ^\ tan 6=- i-- ^, (Art. 16) ^ 1 + tan (9i • tan 62 ^ ' 1 + niiin2 L J If the angle were measured from line (1) to line (2) it would be the negative, or else the supplement, of ; in either case its tangent would be the negative of that given by formula [14]. If the equations of the lines had been given in the form : A^ + ^i^ + <^i = 0, ... (3) and A2X + Boi/ -f- (^2 = 0, . . . (4) A A then mi= — — \ 7712 = — ^^ and formula [14] becomes tan0 = lA^^.4,^L:^A^. . . [15] 1^4i42 A,A2 + B,B2 ^ -' B^B2 EXERCISES Find the tangent of the angle from the first line to the second in each of the following cases, and draw the figures : 1. 3a; -43/ -7 = 0, 2a:-?/ -3 = 0; 2. 5a; + 12 3/ + l=0, a;-2?/ + 6 = 0; 3. 2a: = 3?/-|-9, 6?/ = 4a;+2; a b ah 5. a; cos a -\- y sin a = p, - + ^ = 1. a 6 62. Condition that two lines are parallel or perpendicular. From formula [14] can be seen at once the relations that ; 61-62.] THE STRAIGHT LINE 99 must hold between rrii and mg if tlie lines (1) and (2) (Art. 61) are parallel or perpendicular. If these lines are parallel, then <^ = 0, and therefore tan = 0; hence —^ = 0, 1 + ^17/12 I.e., nii=in29 which is the condition that lines (1) and (2) are parallel.* This condition is also evident from a mere inspection of equations (1) and (2). If the lines (1) and (2) (Art. 61) are perpendicular, then (j) = 90° and tan (f) = cc , ^.e., ; — — = GO , hence 1 + wiimg = 0, 1 + mim2 1 ^.e., fn2 = 9 mi which is the condition that (1) and (2) are perpendicular. So also from [15] the lines Aix + Bii/ + (7^ = and A2X + A«/ + C2 = are parallel if (and only if) ^2^1 — ^1^2 = 0, ^'.e., if Ai :^i = J.2: ^2; and they are perpendicular if (and only if) J.i^2 + -^i^2=0? i.e., if Ai'. Bi=—B2: A- The condition just found enables one to write down readily the equations of lines that are parallel or perpendicular to given lines, and which also pass through given points. * It must not be forgotten that this conclusion is drawn only for lines that are not perpendicular to the a:-axis ; because if the lines are perpen- dicular to the X-axis then equations (1) and (2) are inapplicable (cf. Art. 56). 100 ANALYTIC GEOMETRY [Ch. V. E.g.^ let it be required to write the equation of a line that is parallel to the line y = ^x+'J. . . . (1) The slope of this line is 3, hence any other line whose slope is 3 is parallel to the given line, i.e., 7/ = Sx-\-h, . .- . (2) is, for all values of 5, parallel to line (1). If it be required that the line (2) shall also pass through a given point, (1, 5) for example, it is only necessary to determine rightly the value of b. This is done by remem- bering that if the line (2) passes through the point (1, 5), then these coordinates must satisfy equation (2), i.e., 5 = 3 • 1 4- J, whence 5 = 2. Therefore the line i/ = Sx + 2 is not only parallel to the line ?/ = 3 a; + 7, but also passes through the point (1, 5). Similarly y = — i a; + 6, whatever the value of 5, is per- pendicular to y = Sx-\- 1. Again, the line Sx + 5i/-{-k = 0, whatever the value of k, is parallel to the line 3a: + 5y — 15 = 0; and the line 5x — Sy'{-k = is perpendicular to 3 a; -1- 5 y — 15 = 0. Here again the arbitrary constant k may be so determined that this line shall pass through any given point. So also the lines A^x + B^y + (7^ = and A^x + B^y -{- C^^ — are parallel, while A^x -|- B^y + 6^ = and B^x — A-^y -\- C^ = are perpendicular to each other. This condition for parallelism and for perpendicularity of two lines may also be stated thus : two lines are parallel if their equations differ (or may be made to differ) only in their constant terms ; two lines are 'perpendicular if the coeffi- cients of X and y in the one are equal (or can be made equal), respectively., to the coefficients of —y and x in the other. 62-63.] THE STRAIGHT LINE 101 EXERCISES 1. Write down the equations oi" the set of lines parallel to the lines : (a) y = Qx-2; (f3) 3x-7y = 3; (y) a:cos30° + 2/sin30° = 8; (8) ^-| = 1. 2. Explain why it is that the constant term in the answers to Ex. 1 is left undetermined or arbitrary. 3. Find the tangent of the angle between the lines (a) and ((3) in Ex. 1 ; also for the lines (/3) and (8), and (a) and (8) of Ex. 1. 4. Write the equations of lines perpendicular to those given in Ex. 1. 5. By the method of Art. 62 find the equation of the line that passes through the point ("9, 1), and is parallel to the line y = Qx — 2. 6. Solve Ex. 4 by means of equation [11], Art. 53. 7. Find the equation of the line that is parallel to the line Ax + By + C = and that passes through the point (x^, y^ ; make two solu- tions, one by the method of Ex. 4, and the other by Ex. 5. Find the equation of the straight line 8. through the point (2, —5) and parallel to the line y = 2 x + 1 . 9. through the point ("l, —1) and perpendicular to y = 2 x -\- 7 ] solve by two methods. 10. through the point (0, 0) and parallel to the line 3 7 X - y -^ 1 r-iy= — 9 — 11. perpendicular to the line 2y-\-1x — l = Q, and passing through the point midway between the two points in which this line meets the coordinate axes. 12. Find the foot of the perpendicular from the origin to the line 5x-7?/ = 2. 63. Line which makes a given angle with a given line. The formula , tan 6. — tan 0^ ^ k , n-i^ tan 6 = 3 , ^ ^ 1- (Art. 61) ^ 1 4- tan 6^ tan 0^ ^ ^ states the relation existing between the tangents of the angles 6^^ 0^, and cf) (see Fig. 47) ; hence if any two of these 102 ANALYTIC GEOMETRY [Ch. V. angles are known, this equation determines the value of the third. Thus this formula may be employed to determine the slope of a line that shall make a given angle with a given line. E.g.^ given the line 3?/ — 5a; + 7 = 0, to find the equation of a line that shall make an angle of 60° with this line. Here (j) = 60°, i.e., tan = V3, and if 6^ be the angle which the given line makes with the 2:-axis, and 6^ that made by the line whose equation is sought, then tan ^^ = |-. Substi- tuting these values in the above formula, it becomes whence V3^ I — tan ^2 1 4- 1 tan ^2' , . 5-3V3 , 5-3V3 tan To = — =, and y = z — — 'X -\- k ^ 3 + 5V3 ^ 3 + 5V3 is the equation of a line fulfilling the required conditions, — k may be so determined that this line shall also pass through any given point. It is to be remarked that through any given point there may be drawn two lines, each of which shall make, with a given line, an angle of any desired magnitude. E.g., through P^= {x^, y^ the lines (1) and (2) may be so drawn that each shall make an angle with the given 63.] THE STRAIGHT LINE 103 line LM. Let line (1) make an angle 0^^ line (2) an angle ^2? and LM an angle ^3, with the a;-axis ; then <^ = 6>i - ^3, and 180- = ^2 - ^3 » which gives tan ^1 — tan ^q , , , tan^, — tan^o tan 6 = ^ r-^— 7^ ^, and — tan 6 = z ^ n ^ — n ^ 1 + tan ^1 tan 0^' ^ 1 -f tan 6^ tan d^ In these equations cf) and 0^ are known, hence tan 0^ and tan ^2 ^^^ ^® found. Having found tan^^ and tan ^2 ^^^ equations of lines (1) and (2) may at once be written down, either by means of equation [11], or by the method employed in Art. 62. EXERCISES 1. Find the equations of the two Hnes which pass through the point (5, 8), and each of which makes an angle of 45° with the Ime 2 x — 3y = Q. 2. Show that the equations of the two straight Unes passing through the point (3, -2) and inclined at 60° to the line x V3 + ?/ = 1 are y + 2 =0 and ?/-xV8 + 2 + 3V3 = 0. Find the equation of the straight line 3. making an angle of — — with the line 3 a: — 4 y = 7 ; construct the figure. Why is there an undetermined constant in the resulting equation ? 4. making an angle of + 60° with the line 5x+12y-\-l = 0; con- struct the figure. 5. making an angle of — 30° with the line x — 2y + l=0, and passing through the point (1,3); making an angle of + 30°, and passing- through the same point. 6. making an angle of it 135° with the line x + y = 2, and passing through the origin. J) or 7J 7. making the angle tan"^ - with the line - + ^ = 1, and passing f h\ ^ a b through the point f -, - j . 8. Find the equation of a line through the point (4, 5) forming with the lines 2x — y-\-^ = and 3y + 6a; = 7a right-angled triangle. Find the vertices of the triangle (two solutions). 104 ANALYTIC GEOMETRY [Ch. V. 9. Show that the triangle whose vertices are the points (2, 1), (3, -2), (-4, -1) is a right triangle. 10. Prove analytically that the perpendiculars erected at the middle points of the sides of the triangle, the equations of whose sides are X + y -\-l = 0, ?> X + b y + 11 = 0, and x-\-2y + ^ = 0, meet in a point which is equidistant from the vertices. 11. Find the equations of the lines through the vertices and perpen- dicular to the opposite sides of the triangle in exercise 10. Prove that these lines also meet in a common point. 12. A line passes through the point (2, -3) and is parallel to the line through the two points (4, 7) and (-1,-9) ; find its equation. 13. Find the equation of the line which passes through the point of intersection of the two lines 10a: + 5?/ + ll = 0, and a; + 2?/ + 14 = 0, and which is perpendicular to the line x-f7?/+l = 0. This problem may be solved by first finding the point of intersection i^i — ^) of the two given lines, and then, by formula [11] (see also Art. 62), writing the equation of the required line, viz. : 2/ + ^ = 7 (:r - ^), which reduces to 7 a: — ?/ = 31. The problem may also be solved somewhat more briefly, and much more elegantly, by employing the theorem of Art, 41. By this theorem the equation of the required line is of the form 10 a: + 5 ?/ + 11 + A' (a: + 2 ?/ + 14) = 0, i.e., (10 + ;^) a: + (5 + 2 yt) !/ + 11 + 14 ^ = 0. It only remains to determine the constant h, so that this line shall be perpendicular to a: + 7 ?/ + 1 = 0. By Art. 62 its slope must be = 7, hence — — = 7, whence ^ = — 3. Substituting this value of Tc above, the required equation becomes 7 a: — 2/ = 31, as before. 14. By the second method of exercise 13 find the equation of the line which passes through the point of intersection of the two lines 2 a: + y = 5 and a; = 3y — 8, and which is : (1) parallel to the line 4?/ = 6 a: + 1 ; (2) perpendicular to this line ; (3) inclined at an angle of 60° to this line ; (4) passes through the point ("1, 3). 15. Solve exercise 10 by the method of exercise 14. 63-64.] THE STBAIGHT LINE 105 16. Do the lines 2 a; + 3 y = 13, 5x - y = 7, and a:-4?/ + 10 = meet in a common point ? What are the angles they make with each other ? 17. Find the angles of the triangle of exercise 10. 18. When are the lines x-{-(a-{-h)y + c = and a(x + ay) -^ h (x — by) + d ■= parallel? when perpendicular? 19. Find the value oi p for each of the two parallel lines y = 3 X -\- 7 and y = S x — 5] and hence find the distance between these lines [cf. Art. 58 (3) and (4)]. 20. What is the distance between the two parallel lines 5x -3y + 6 =0 and Qy - 10a; = 7? 21. Find the cosine of the angle between the lines y - 4:x + 8 = and y-6x + 9 = 0. 22. What relation exists between the two lines y = 3x + 7 and y = - 3 a: - 3 ? 23. Find the angle between the two straight lines 3 a: = 4 ?/ + 7 and 5y = 12a; + 6; and also the equations of the two straight lines which pass through the point (4, 5) and make equal angles with the two given lines. 24. Find the angle between the two lines 3x + y + 12 = and x + 2y-l = 0. Find also the coordinates of their point of intersection, and the equations of the lines drawn perpendicular to them from the point (3, ~2). 64. The distance of a given point from a given line. This problem is easily solved for any particular case thus : find the equation of the line which passes through the given point and which is parallel to the given line (Art. 62), then find the distance (jt?) from the origin to each of these two lines [Art. 58, (3) and (4)], and finally subtract one of these distances from the other ; the result is the distance between the given line and the given point. 106 ANALYTIC GEOMETRY [Ch. V. Fig. 49. E.g.^ find the distance of the point Pi=(2, |) from the ^^^^ 3a;4-i?/-7 = 0. . . . (1) Let line (1) be the locns of equation (1), andPj be the given point. Through Pj draw the line (2) par- allel to line (1), also draw QP^ perpendicular to line (1), OE^( = p^) perpen- dicular to line (1), and (9i?2( =^2) perpendicular to line (2) . Then d = QP^ =p^ —py The equation of a line parallel to line (1) is of the form 3a: + 'iz/-f^=0; this will represent line (2) itself if k be so determined that the line shall j)ass through the point Pj=(2, I), i.e,,ii 3-2h-4.| + A: = 0, i.e., if ^ = - 12. The equation of line (2) is then 3a; + 4z/-12 = Therefore [by Art. 58, (3) or (4)] 12 12 , 7 7 — . and r>. = (2) P2 = 4-V4^ + 3-^ 12 = = -, and;>i = + V42 + 32 5' hence the required distance is t?= QP^ = 12-7 = 1. Similarly, in general, to find the distance of any given point P^ = (a;^, ^/j) from any given line Ax-hBi/-\-C=0 (1) let line (1) be the locus of equation (1) and let P^ be the given point. The equation of a line parallel to (1) is of the form Ax ■{- Bi/ -\- K=0 ; this will be the line (2) if 64.] THE STRAIGHT LINE 107 Ax^ + %i + K= 0, i.e., if K= - (Ax^ + Bi/^). The equa- tion of line (2) is then Ax-\-Bi/-(Ax^i-Bi/^)=0. ... (2) Therefore /?„ = — ^i "^ '^ ^ , » = — ~ wherein the sign of the radical is to be chosen in accord with Art. 58 (3); hence ^^ Ax,+Bp,^C ^ ^ ^ ^ .^g-. If the equation of the given line is so written that its second member is zero, this formula may be translated into words thus : To get the distance of a given point from a given line, write the first member of the equation alone, substitute for the variables therein the coordinates of the given point, and divide the result by the square root of the sum of the squares of the coefficients of x and y in the equation, — the sign of this square root being chosen opposite to that of the number represented by C. If, in formula [16], d is positive, then p^> Pi, and P^ and the origin are on opposite sides of the given line ; if d is negative, P2, ... (1) and 12a;-53/ + 6 = 0. . . . (2) 64-65.] THE STRAIGHT LINE 109 Let Pj = (a;j, ?/j) be any point on the bisector (3). Then Q^P^ = -R^P^ [since and P^ are on opposite sides of line (1) and On the same side of (2) ; or vice versa]. But np 3»,+4.y^ -l + V32 + 42 = ^^^ + \y,-\ (Art. 64), Y / \- .R, r "--^.^ (3) A \ X / \ ^ X m ^ Vi) \a) Fig, 50. and jip ^ 12 rr^ - 5 ,y, + 6 ^ 12 rr, - 5 ,yT + 6 . ^ ^ _Vl22-i-.^2 -13 Hence (5) (6) VI22 + 52 . 3a;^ + 4yT-l _ 12:ri-5,y, +6 . 5 " 13 t.e., 21 0^1 - 77 ?/i + 43 = 0. . . , 21 2; -77?/ + 43 = . . . is the equation of the bisector (3), for equation (5) asserts that if (a^j, 1/1) be the coordinates of any point on this bisec- tor they satisfy equation (6). Similarly, let P^ = (^, k) be any point on line (4), the other bisector, then $2^2 — -^2^2 [since and P^ are on opposite sides of the lines (1) and (2), or else both on the same side of each of these lines] ; Sh-h4:k-l 12h-bk-h6 5 ~ 13 ' 99A + 27Aj + 17 = 0. . . . (7) 99a: + 27y + 17 = ... (8) is the equation of the bisector (4), for the same reason as given above. I.e., Hence 110 ANALYTIC GEOMETBY [Ch. V. Geometrically it is well known that two such bisectors, (3) and (4), are perj^endicular to each other : their equa- tions, also prove that fact. The equations of the bisectors of the angles between any two lines, as A^x + B-^y + (7^ = and A^^x + B^y + C\ = 0, are found in precisely the same way as that employed in the numerical example just considered. EXERCISES 1. Find the equations of the bisectors of the angles between the two lines X — y + Q = and — ^ — = oy —7. 2. Show that the line llx-^^y + l = bisects one of the angles between the two lines 12 x - 5 y + 7 = 0, and 3 x -^ 4: y - 2 = 0. Which angle is it ? Find the equation of the bisector of the other angle. 3. Show analytically that the bisectors of the interior angles of the triangle whose vertices are the points (1, 2), (5, 3), and (4, 7) meet in a common point. 4. Show analytically, for the triangle of Ex. 3, that the bisectors of one interior and the two opposite exterior angles meet in a common point. 5. Find the angle from the line 3a:4-2/ + 12=:0to the line ax + by + 1 = 0, and also the angle from the line ax -\- by + 1 = to the line X + 2y -1 =0. By imposing upon a and b the two conditions : (1) that the angles just found are equal, and (2) that the line ax -\- by -}- 1 = passes through the intersection of the other two lines, determine a and b so that this line shall be a bisector of one of the angles made by the other two given lines. 66. The equation of two lines. By the reasoning given in Art. 40, it is shown that if two straight lines are represented by the equations . A^x-hB^y^C^ = , . . (1) and A^x + B^y + (72 = 0, . . . (2) then both these lines are represented by the equation iA^x + B^y + C{)(iA^x + B^ + C^) = 0; . . . (3) 65-67.] THE 8TBA1GHT LINE 111 ^.e., two straight lines are here represented by an equation of the second degree. Conversely, if an equation of the second degree, whose second member is zero, can have its first member separated into two first degree factors, with real coefficients, as in equation (3), then its locus consists of two straight lines. Thus the equation may be written in the form (2a;-3^ + 7)(a;+2/-f-l) = 0, which shows that it is satisfied when 2a; — 3?/+7 = 0, and also when x -\- y -\- 1 = 0. Its locus is therefore composed of the two lines whose equations are : 2a;-3y + 7 = 0, and x + y -[-1 = 0. 67. Condition that the general quadratic expression may be factored. The most general equation of the second degree between two variables may be written in the form Ax^+2Exy^-By'^+'iax-^2Fy +0=0. , . . (1) It is required to find the relation that must exist among the coefficients of this equation in order that its first member may be separated into two rational factors, each of the first degree, i.e., it is required to find the condition that the equa- tion may be written thus : (^ct^x -f h^y + c^{a^x + h^y -{-c^^=0. . . . (2) Evidently if equation (1) can be written in the form of equation (2), then the values of x obtained from equation (1) are rational, and are either ^^-^^-hy or a; = - ^2 - hy, H «2 112 ANALYTIC GEOMETRY [Ch. V. Solving equation (1) for x in terms of ?/, by completing the square of the 2;-terms, it becomes . ^v + 2 ^(% + a)x + {iTy + ay = - ABf-2AFi/ -AQ + {Hy + ay, i.e., Ax-\-Hy+ a = V(^2 _ AB)y'^ - 2 {Ha - AF)y + a^-AO, and finally, H a , 1 x=---y--±--VCE'^-AB)y''-2{Ra-AF}y+a^-A0. A Jx JL But since x is, by hypothesis, expressible rationally in terms of y, therefore the expression under the radical sign is a perfect square, and therefore CHa - AFy-^H'^- AB)(a^ - ao)= o, i.e., ABC + 2 FGH- AF^ - BG^ - CH^ = 0. . . [17] If this condition among the coefficients is fulfilled, then equation (1) has for its locus two straight lines. The expression AB0+2FaR - AF^ - Ba'^ - CR^ is called the discriminant of the quadratic, and is usually represented by the symbol A. Note. The analytic work just given fails if ^ = 0. In that case equation (1) may be solved for y instead of solving it for x, and the same condition, viz. A = 0, results. If, however, both A and B are zero, then the above method fails altogether. In that case equation (1) reduces to 2Hxy + 2Gx + 2Fy+C = (3) If the first member of equation (3) can be factored, then evidently the equation must take the form {ax + b)(cy + d) = Q ....... (4) which shows that equation (3) is satisfied for all values of y provided X = — , a constant. Let be represented by k, then equation (4) a a becomes 2 Hky + 2 Gk + 2 Fy + C = 0, i.e., 2iHk + F)y + 2Gk+C = 0, 67.] THE STRAIGHT LINE 113 and is satisfied for all values of y ; Hk-^ F=0, and 2Gk-{-C = 0; hence, eliminating k, 2 FG — CH = 0. But this is the expression to which A reduces when A = B = and H =^0] hence, in all cases, A = is the necessary condition that the above quadratic may be factored. That A = is also the sufficient condition is readily seen by retracing the steps from equation [17] when at least one of the coefficients .4, B differs from zero. But it is also sufficient when ^4 = B = 0] for, in that F G C case, A = becomes 2 FG — CH = 0, which may be written — . — = — — . X H H 2 H Under the same circumstances equation (1) becomes equation (3), which may be written xy+^xi-^y+-^ = (4) -^ H H^ 2H ^ ^ F G G Substituting for in equation (4), it becomes H H 2 H 3:y + ^x + ^V + ^-^ = (5) ^ H H^ H H ^ ^ (. + |)(. + |)=o, which establishes the sufficiency of the condition for this case also. To illustrate the use of equation [17]* examine the equation of Art. 66 : 2x^ - xy -oy^ + Qx + ^y + 1 =0. * As an illustration of another practical method of factoring a quadratic expression, lohen factoring is possible-, i.e., if equation [17] holds, find the factors of 2x^ -Ixy -15y'^ + 7x-l'ly -4t. Factor the terms free from y, 2a;2 + 7x-4=(2a;-l)(a; + 4); factor the terms free from x, - 152/2 - 17 y - 4= (3 ?/ - 1)(_ 5y + 4); combine the factors containing the same constant term, (2x + Sy-l), (x-6y+4:); these will be the factors of the given quadratic expression. TAN. AN. GEOM. — 8 114 ANALYTIC GEOMETRY [Ch. V. Here ^ = 2, 5 = - 3, C = 7, i^ = - |, G=|,andF = 2; hence a = - 42 - 9 - 8 +^ - 7 = 0; 4 4 therefore the first member can be factored. The factors may be found as follows : transposing, dividing by 2, and completing the square of the a;-terms, the equation may be written in the form x^ + ^^ + (^)' "^ll^^' - 2^/ + 1); therefore the given equation, divided by 2, may be written in the form, i.e., (a; + 3/ + 1) (a; - I ?/ + = ; hence the locus of the original equation consists of the straight lines a: + ?/ + 1 = and 2a;-33/ + 7 = 0, which agrees with the result of Art. 66. EXERCISES Prove that the following equations represent pairs of straight lines ; find in each case the equations of the two lines, the coordinates of their point of intersection, and the angle between them. 1. 6 y2 _ x^ - a:2 -f 30 ?/ + 36 = 0. 2. a:2-2a:?/-3 3/2 + 2a;-23/ + l=0. 3. x^ — 2 x?/ sec a + 2/2 = 0. 4. a;2 + 6 a;?/ + 9 ?/2 + 4 a: + 12 ?/ - 5 = 0. 5. For what value of k will the equation a:2 _ 3 ^^ ^ 2/2 + 10 a: - 10 2/ + Z: = represent two straight lines ? Suggestion : Place the discriminant (A) equal to zero, and thus find A; = 20. Find the values of k for which the following equations represent paii-s of straight lines. Find also the equation of each line, the point of inter- section of each pair of lines, and the angle between them. 6. 6 a;2 + 2 kxy + 12 ?/2 + 22 a: + 31 ?/ + 20 = 0. 7. 12 a;2 + 36 a;?/ + A;2/2 + 6 a; + 6 ^ + 3 = 0. 67-68.] THE STRAIGHT LINE 115 8. 4iX^-12xy ^Qy"^ - kx -{-6y +1 = 0. 9. The equations of the opposite sides of a parallelogram are a:2 _ 7 a; + 6 = and 1/2 - 14 2/ + 40 = 0. Find the equations of the diagonals. 10. Find the conditions that the straight lines represented by the equa- tion Ax"^ + 2 Bxy + Cy^ = may be real ; imaginary; coincident ; perpen- dicular to each other. 11. Show that 6 x2 + 5x2/ - 6 2/2 = is the equation of the bisectors of the angles made by the lines 2 x^ + 12 xy + 7 y"^ = 0. Does the first set of lines fulfil the test of exercise 10 for perpendicularity? 68. Equations of straight lines : coordinate axes oblique. Since in the derivation of equations [9] and [10] (Arts. 51 and 52) only properties of similar triangles were employed, therefore these two equations are true whether the coordi- nate axes are rectangular or oblique. The other three standard forms however, viz. ?/ = mx + 5, y — y^ = m(x —x{)^ and x cos a +?/ sin a=p^ the derivation of which depends upon right triangles, are no longer true if the axes are inclined to each other at an angle « ^ ^- Equa- tions which correspond to these, but which are referred to oblique axes, will now be derived. (1) Equation of straight line through a given point and in a given direction. Let LL^ be the straight line through the fixed point P^=(x^^ y^ and making an angle 6 with the ic-axis, let P = (a;, y^ be any other point on XZ/j, and let CO be the angle between the axes. Draw P^R parallel to the a;-axis, also draw the ordinates M^P^ and MP. Then 6=ZXAL=:ZRP^L and ZP^PR=(o-6. 116 ANALYTIC GEOMETRY [Ch. V. Hence -— -^ = -r— 7 -xr- [law of smesj Pji^ sin(ft)-^) Substituting in this equation the coordinates of P^ and P, it becomes y - yi ^ sin ^ 0^ — a?! sin (w — ^)' sin Or ^ n Qi which is the required equation. When o) = - this equation reduces to equation [11], ^.e., to y — yi= 'rn {x — a^i), where m = tan 6 ; but it must be observed that if « =?^^, then the coefficient of x in equation [18] does not represent the slope of the line. If, however, the slope of the line [18], i.e.^ the tan 6 for this line, is desired, it is easily found thus : let = ^, from ^ sin (a)-6>) which is obtained tan 6 = . 1 -\- k cos (o If, in the derivation of equation [18], the given point is that in which the line LL-^ meets the 2/-axis, ^.e., if Pi =(0, i), then equation [18] reduces to y= . 7^ .. ^ + h, ■ • • [19] sm (o) — 6) which corresponds to equation [12], but the coefficient of x is not the slope of the line. (2) Equation of a straight line in terms of the perpendic- ular upon it from the origin, and the angles which this perpe7i- dicular inakes with the axes. 68.] THE STRAIGHT LINE 117 Let LL^ be the straight line whose equation is sought, let the perpendicular from the origin upon it (01^= p) make the angles a and ^ respectively with the axes,* and let P= (a:, «/) be any point on LLi. Draw the ordinate MP ; then, / fig.52. by Art. 17, OM cos a + MP cos /3 = ON, i.e.^ X cos a -\- y cos ^ = p^ . . . [20] which is the required equation. If ft) is the angle between the axes, then /3 = &> — a, and equation [20] may be written x cos a ■\- y cos (o) — a) =Jt?. If ft) = - , then this equation reduces to x cos a -Vy sin a-= p, Li which agrees with equation [13]. EXERCISES 1. The axes being inclined at the angle 60°, find the inclination of the line ?/ = 2 a: + 5 to the a:-axis. 2. The axes being inclined at the angle -, find the angles at which the lines 32/ + 7a:-l=0 and a: + ?/ + 2 = cross the a:-axis. 3. Find the angle between the lines in exercise 2. 4. The center of an equilateral triangle of side 6 is joined by straight lines to the vertices. If two of these lines are taken as coordinate axes, find the coordinates of the vertices, and the equations of the sides. 5. Prove that for every value of w, the lines a; + 2/ = c and x — y = d are perpendicular to each other. * The angles a and j3 are the direction angles of the line ON, and their cosines are the direction cosines of that line. 118 ANALYTIC GEOMETRY [Ch. V. 69. Equations of straight lines: polar coordinates. (1) Line through two given points. Let OR be the initial line, the pole, Pi = (pi, ^i)^ and P2 = (/02, 62)-, the two given points, and let P = (/3, 6) be any other point on the line through Pi and Pg. Fig. 53. Then (if A stands for ' area of triangle ') A OPiP, = A OPiP + A OPP2. i.e., I piP2sin(^2-^i) = J/o/?isin(<^-<^i)+ I P2P sin (^2-^)^ hence ppi sin (^— ^1) 4-/3ip2 sin (^1 — ^2) + /92psin(i92-6>) = 0.*. . . [21] This equation may also be written in the form sin {Oi - 62) I sin (jO^ - 6) ^ sin {0 - Oi) ^^ ,, P Pi P2 (2) Equation of the line in terms of the perpendicular upon it from the jjole^ and the angle which this perpendicular makes with the initial line. Let OR be the initial line, the pole, and LK the line whose equation is sought. Also, let N= (p, a) be the foot of the perpendicular from upon LK., and let P=(p., 6) be any other point on LK. Draw ON and OP ; then ON ^^= cos NOP, i.e., p cos (^ — a) = p, which is the required equation. [22] * Observe the symmetry here ; cf . foot-note, Art. 29. 69.] THE STRAIGHT LINE 119 EXERCISES 1. Construct the lines : (a) p cos ((9- 30°) = 10; (c) pcos^^-|^=9; (&) p sin = 2', (d) p cos {6 - tt) = Q. 2. Find the polar equations of straight lines at a distance 3 from the pole, and: (1) parallel to the initial line; (2) perpendicular to the initial line. 3. A straight line passes through the points (5, -45°) and (2, 90°) ; find its polar equation. 4. Find the polar equation of a line passing through a given point (/Op Oi) and cutting the initial line at a given angle = t3in~'^k. 5. Find the polar coordinates of the point- of intersection of the lines p cos ( ^ — ^ J = 2 a, p cos ( ^ - ^ J = a. EXAMPLES ON CHAPTER V 1. The points ("1, 2) and (3, -2) are the extremities of the base of an equilateral triangle. Find the equations of the sides, and the coordi- nates of the third vertex. Two solutions. 2. Three of the vertices of a parallelogram are at the points (1, 1), (3, 4), and (5, -2). Find the fourth vertex. (Three solutions.) Find also the area of the parallelogram. 3. Find the equations of the two lines drawn through the point (0, 3), such that the perpendiculars let fall from the point (6, 6) upon them are each of length 3. 4. Perpendiculars are let fall from the point (5, 0) upon the sides of the triangle whose vertices are at the points (4, 3), (-4, 3), and (0, ~5). Show that the feet of these three perpendiculars lie on a straight line. Find the equation of the straight line 5. through the origin and the point of intersection of the lines X — y = 4: and 7 x -{- y -]- 20 = 0. Prove that it is a bisector of the angle formed by the two given lines. 6. through the intersection of the lines 3x — 4?/ + l=0 and 5x + y = 1, and cutting off equal intercepts from the axes. 7. through the point (1, 2), and intersecting the line a; + y = 4 at a distance | V6 from this point. 120 ANALYTIC GEOMETRY [Ch. V. 8. Find the equation of a straight line through the point (4, 5) and making equal angles with the lines 3 x = 4 ?/ + 7 and by = 12x -\- Q. 9. Prove analytically that the diagonals of a square are of equal length, bisect each other, and are at right angles. 10. Prove analytically that the line joining the middle points of two sides of a triangle is parallel to the third side and equal to half its length. 11. Find the locus of the vertex of a triangle whose base is 2 a and the difference of the squares of whose sides is 4 c^. Trace the locus. 12. Find the equations of the lines from the vertex (4, 3) of the tri- angle of Ex. 4, trisecting the opposite side. What are the ratios of the areas of the resulting triangles ? 13. A point moves so that the sum of its distances from the lines ^-3x4-11 = and 7 x -2y + I = is 6. Find the equation of its locus. Draw the figure. 14. Find the equation of the path of the moving point of Ex. 13, if the distances from the fixed lines are in the ratio 3 : 4. 15. Solve examples 13 and 14, taking the given lines as axes. 16. The point (2, 9) is the vertex of an isosceles right triangle whose hypotenuse is the line dx — 7 y = 2. Find the other vertices of the triangle. 17. The axes of coordinates being inclined at the angle 60°, find the equation of a line parallel to the line x -^ y = 3 a, and at a distance aV3 ^ ^2 from it. 18. Find the point of intersection of the lines P = -P :- and pcos(^--) = a. cos a — For what value of &, in each line, is p = oo ? At what angles do these lines cut their polar axes? Find the angle between the lines. Plot these lines. 19. Find the equation of a straight line through the intersection of y — 7x — 4: and 2x -\- y = 5, and forming with the x-axis the angle -• 20. Find the equation of the locus of a point which moves so as to be always equidistant from the points (2, 1) and (~3, ~2). 69.] THE STRAIGHT LINE 121 21. Find the equation of the locus of a point which moves so as to be always equidistant from the points (0, 0) and (3, 2). Show that the points (0, 0), (3, 2), and (1, -1) are the vertices of an isosceles triangle. 22. Find the center and radius of the circle circumscribed about the triangle whose vertices are the points (2, 1), (3, -2), (-4, -1). 23. Find analytically the equation of the locus of the vertex of a triangle having its base and area constant. 24. Prove analytically that the locus of a point equidistant from two given points (x^, y^ and {x^, 2/2) is the perpendicular bisector of the line joining the given points. 25. The base of a triangle is of length 5, and is given in position ; the difference of the squares of the other two sides is 7 ; find the equa- tion of the locus of its vertex. 26. AVhat lines are represented by the equations : (a) x'^y = xy^ ; (^) 14 x^ — 5 a:?/ — y^ = ; (y) xy = 0? 27. What must be the value of c in order that the lines 3x-{-y — 2 = 0, 2a:— !/ — 3 = 0, and 5x + 2y + c = shall pass through a common point ? 28. By finding the area of the triangle formed by the three points (3 a, 0), (0, 3 6) and (a, 2 b), prove that these three points are in a straight line. Prove this also by showing that the third point is on the line join- ing the other two. 29. Find, by the method of Art. 39, the point of intersection of the two lines 2 x — Sy -{-7 = and 4 a: = 6 ?/ + 2 ; and interpret the result by means of Arts. 41 and 60. 30. Prove by Art. 10 (cf. also Arts. 41 and 60), that the equations of two parallel lines differ only in the constant term. 31. Find the equations of two lines each drawn through the point (4, 3), and forming with the axes a triangle whose area is 8. 32. Find the equation of a line through the point (2, -5), such that the portion between the axes is divided by the given point in the ratio 7:5. 33. Find the equation of the perpendicular erected at the middle point of the line joining (5, 2) to the intersection of the two lines x + 2y = ll and 9x-2y=59. 122 ANALYTIC GEOMETRY [Ch. V. 69. 34. A point moves so that the square of its distance from the origin equals twice the square of its distg^nce from the x-axis ; find the equation of its locus. 35. Given the four lines x-2y + 2 = 0, X + 2ij -2 = 0, dx-y -S = and a; + ?/+6 = 0; these lines intersect each other in six points ; find the equations of the three new lines (diagonals), each of which is determined by a pair of the above six points of intersection. 36. Find the points of intersection of the loci : (a) p cos ( ^ - ^ J = a and p cosf ^ - ^ j = a ; (13) pcos(0-f\ = ^ Siud p = asme. If two sides of a triangle are taken as axes, the vertices are (0, 0), (Xj, 0), (0, 2/2)- Prove analytically that: 37. the medians of a triangle meet in a point ; 38. the perpendicular from each vertex to the opposite sides meet in a point; 39. the line joining the middle points of two sides of a triangle is parallel to the third side. 40. Show that the equation 56 x^ - 441 xy - oQy^ -79 x - 47 ?/ + 9 = represents the bisectors of the angles between the straight lines repre- sented by 15x2- 16a:?/ -482/2- 2a; + 16?/- 1=0. 41. Two lines are represented by the equation .4x2 + 2 Hxy + %2 = 0. Find the angle between them. CHAPTER VI TRANSFORMATION OF COORDINATES 70. That the coordiuates of a point which remains fixed in a plane are changed by changing the axes to which this fixed point is referred, is an immediate consequence of the definition of coordi- nates. It is also evident that the different kinds of coordinates of any given point (Cartesian and polar, for example) are connected by definite relations if the ele- ments of reference (the axes) are related in position. E.g.i, the point §, when referred to the polar axis OX and the pole 0, has the coordinates (5, 30°), but when it is referred to the rectangular axes OX and 0!F the coordinates of this same point are (f V3, |) ; and gen- erally, if (/?, ^) be the co- ordinates of a point when referred to OX and 0, then (/o cos ^, p sin 0') are its coordi- nates when it is referred to the rectangular axes OX and OY. Again : while a curve remains fixed in a plane, its equa- tion may often be greatly simplified by a judicious change of 123 r .0' X Fig. 56 124 ANALYTIC GEOMETRY [Ch. VI. the axes to which it is referred, ^.g., the line L^L^ when referred to the axes OX and OY, has the equation «/ = tan 6 -x + b, but when referred to the axes O'X' and 0' Y'^ the former of which is parallel to the given line, its equation is ?/ = c. For these, and other reasons, in the study of curves and surfaces by the methods of analytic geometry, it will often be found advantageous to transform the equations from one set of axes to another. It will be found that the coordinates of a point with reference to any given axes, are always connected by simple formulas with the coordinates of the same point when it is referred to any other axes. These relations or formulas for the various changes of axes are derived in the next few articles. I. CARTESIAN COORDINATES ONLY 71. Change of origin, new axes parallel respectively to the original axes. Let OX and Oy be the original axes, 0' X' and O'Y' the new axes, and let the coordinates of the new origin when referred to the original axes be h and h, ^.e., 0' = (A, A:), where h = OA and k = AO'. Also let P, any point of the plane, have the coordi- nates X and y when it is referred to the axes OX and OF, and x' and y' when it is referred to the axes 0' X' and 0' Y' . Draw MM'P parallel to the ?/-axis ; then 0M= OA + AM= OA + O'M', and similarly, y = y' + k. Fig. 57 [23] 70-71.] TRANSFORMATION OF COORDINATES 125 which are the equations (or formulas) of transformation from any given axes to new axes which are respectively parallel to the original ones, the new origin being the point 0' = (h^k^. These formulas, moreover, are independent of the angle between the axes. As a simple illustration of the usefulness of such a change of axes, suppose the equation x^ - 2hx -\- 1/ - 2ky = a^ - h? - k'^ . . (1) given, in which x and y are coordinates referred to the axes OX and or. Now let F = (2;, ?/) be any point on the locus L^L of this equation, and let (a;', ?/') be the coordinates of the same point P when it is referred to the axes O'X' and 0' Y' ; then x—x'-\-h and y — y' -{-k. Substituting these values in the given equation for the X and y there involved, an equation in x' and y' is obtained which is satisfied by the coordinates of every point on X^i-, z.e., it is the equation of the same locus. The substitution gives : (^' + A)2 -2h(x' -h h^ + iy' + ky -2k(iy'-irk^ = a?-¥-k\ which reduces to a much simpler equation than (1), but representing the same locus, merely referred to other axes. EXERCISES 1. AVhat is the equation for the locus of 3 a: — 2 ?/ = 6, if the origin be changed to the point (4, 3), — directions of axes unchanged ? 2. What does the equation x^ + y" — ix - 6 y = IS become if the origin be changed to the point (2, 3), — directions of axes unchanged? 126 ANALYTIC GEOMETRY [Ch. VI. 3. What does the equation y'^ — 2x^ — 27/ + 6x — 3 = become ^Yhen the origin is removed to (f, 1), — directions of axes unchanged? 4. Find the equation for the straight line y = 3x -h I when the origin is removed to the point (1, 4), — directions of axes unchanged. 5. Construct appropriate figures for exercises 1 and 4. 72. Transformation from one system of rectangular axes to another system, also rectangular, and having the same origin : change of direction of axes. Let OX and OY he sl given pair of rectangular axes, and let OX' and OF' be a second pair, with Z.XOX' = ^, the angle through which the first pair of axes must be turned to come into coincidence with the second. Also let P, any point in the plane, have the coordinates x and y when it is referred to the first pair of axes, and x' and y' when referred to the second pair. The problem now is to express x and y in terms of x\ y\ and 6. Draw the or- dinates MP, M'P, and QM', and draw M'B parallel to the ir-axis; then OM = OQ + QM= OM' cos 6 - M'F sin 6*, i.e., a? = a?' cos 6 - 2/' sine, 1 • • • [24] and similarly, y = oc' sinQ -]- y cos 0, J which are the required formulas of transformation from one pair of rectangular axes to another, having the same origin but making an angle 6 with the first pair. Note 1. These formulas are more easily obtained, — in fact, they can be read directly from the figure, — if one recalls Art. 17, and considers that the projection of OP equals the projection of OM' + the projection of M'P, upon OX and OF in turn. ISToTE 2. The reader will observe that a combination of the trans- formation of Art. 71 with that of Art. 72 will transform from one pah of rectangular axes to any other pair of rectangular axes. 72-73.] TRANSFORMATION OF COORDINATES 127 EXERCISES Turn the axes through an angle of 45°, and find the new equations for the following loci : 1. x'' + y^ = 16', 2. a;2-3/2 = i6; 3. ^ ^ ^^ _ 1 . 4. 17 a;^- 16 a:^ + 17 2/2 = 225. 5. If the axes are turned through the angle tan-i2, what does the equation 4:xy — 3x^ = a^ become ? 73. Transformation from rectangular to oblique axes, origin unchanged. Let OX and OYhe a> given pair of rectangular axes, let OX' and OY' be the new axes making an an- gle ft) with each other, and let the angles XOX' and XOY' be denoted by d and <^, respectively. Also let P, any point in the plane, have the coordinates X and y when referred to the first pair of axes, and x' and y' when referred to the second pair. Draw the ordinates MP, M' P, and QM', also draw M'B parallel to the rr-axis. Then 0M= OQ-hQM= OM' cos6' + MP sin (90 - <^); i.e., x = x' cos -\- y' cos , 1 * and similarly, y = ^' sin -\- y' sin (/>, J which are the required formulas of transformation from rectangular to oblique axes having the same origin. If ft) = 90°, and consequently cf) = 90*^ + 6, then formulas [25] reduce to [24], and Art. 73, therefore, includes Art. 72 as a special case. By first solving for x' and y\ formulas [25] may also be employed to transform from oblique to rectangular axes. Y / / / /90°4

, (d\ 0^ and (/>, respectively. Also let P, any point in the plane, have the coordinates x and y when referred to the first pair of axes, and x' and y' when referred to the second pair. Draw M'P parallel to OF', MP and QM' parallel to OY, and M' R parallel to OX. Then, from the triangle OQM', sin 6 sm o) sin (o and from the triangle RM'P, 0$ = ^'?ijli^J^ and(?M' = BM' = y' "'"(-^ - ") and RP = y' ?Hli- sin © sin o) But 0M= OQ- RM', and MP = QM' + RP ; 73-75.] TRANSFORMATION OF COORDINATES 129 ... ,^-^^ sm(ft)-6>) ^ ^, sm(ft)-(^) ^ sill (0 sm CO , 1 I sin , .sin 6 and y = x' h y' ^- siii ft) sin ft) . . . [26] which are the required formulas of transformation from one pair of oblique axes to another having the same origin. Note. If it is desired to change the origin, and also the direction of the axes, the necessary formulas may be obtained by combining Art. 71 with Art. 72, Art. 73, or Art. 74, depending upon the given and required axes. EXERCISES 1. Show, by specializing some of the angles w, to', 9, and ^ in Art. 74, that formulas [26] include both [25] and [24] as special cases. 2. The equation of a certain locus, when referred to a pair of axes that are inclined to each other at an angle of 60°, is7 x^ — 2xy + 4:y'^ = 5 ; what will this equation become if the axes are each turned through an angle of 30°? What if the x-axis is turned through the angle —30° while the ^-axis is turned through + 30° ? 75. The degree of an equation in Cartesian coordinates is not changed by transformation to other axes. Every formula of transformation obtained ([23] to [26]) has replaced rr aiid y, respectively, by expressions of the first degree in the new coordinates a?', y'. Therefore any one of these transforma- tions replaces the terms containing x and ^ by expressions of the same degree, and so cannot raise the degree of the given equation. Neither can any one of these transforma- tions lower the degree of the given equation ; for if it did, * These formulas can also be read directly from Fig. 60 by first project- ing OM and then the broken line OM'PM upon a line perpendicular to OY ; and afterwards projecting MP and also the broken line MOM'P upon a per- pendicular to OX. The results being equated in each case, and divided by sin w, give [26]. TAN. AN. GEOM. 9 130 ANALYTIC GEOMETRY [Ch. VI. then a transformation back to the original axes (which must give again the original equation) would raise the degree, which has just been shown to be impossible ; hence all these transformations leave the degree of an equation unchanged. II. POLAR COORDINATES 76. Transformations between polar and rectangular sys- tems. (1) Transfoi^mation from a rectangular to a polar system^ and vice versa^ the origin and X-axis coinciding res2:)ectively with the pole and the initial line. Let OX and 1^ be a given set of rectangular axes, and let OX and be the initial line and pole for the system of polar coordinates. Also let P, any point in the plane, have the coordinates x and y when referred to the rectangular axes, and p and 6 in the polar system (Fig. 61), then Fig. 61. similarly, 0M= OP cos XOF; a? = P cos ; 1/ = P sin 9. [27] These are the required formulas of transformation when, hut only when, the rectangular and polar axes are related as above described. Conversely, from formulas [27], or directly from Fig. 61, p = Va?^ + y^, cos 6 = X ^Q^ + ?/2 , and sin 6 y Va;^ + y'^ [28] which are the required formulas of transformation from polar to. rectangular axes, under the above conditions. 75-76.] TRANSFORMATION OF COORDINATES 131 (2) jSame as (1) except that the initial line OR makes an angle cc with the x-axis. It is at once evident that the formulas of transformation for this case are ; X = p cos(^ + «), 1 and y = p sin (^ + a). J The converse formulas for this case are : p = -\/x^ -\- y^ and Q = cos"^ X Vx^ + y^. — a = sni -1 y Vx^ + y' -a. [30] (3) Transformation from any Cartesian system to any polar system. Transform first to rectangular axes whose origin is the proposed pole ; this is accomplished by Arts. 71 and 73. Then by formula [27] or [29] transform from the rectangular Cartesian to the polar coordinates. EXERCISES Change the following to the corresponding polar equations; draw a figure showing the two related systems of axes in each case. Take the pole at the origin, the polar axis coincident with the axis of x, in exercises 1 to 4. 1. x^ -{■ y^ = a' 3. a;2 + 2/2 = 9 (x^ - y^). 2. y^ — X + 2ay = 0. 4. y = x tan a. ,5. X — VSz/ + 2 = 0, taking pole at origin, polar axis making the angle 60° with the x-axis. 6. x'^ — y^ — 4:x — Qy — 54 = 0, taking the pole at the point (2, -3), and the polar axis parallel to the x-axis. Change the following to corresponding rectangular equations. Take the origin at the pole and the x-axis coincident with the polar axis. 7. p = a. 9. p2 sin 2 ^ = 10. 8. p2cos2^ = a2. 10. p2 = a2sin2^. Suggestion. In Ex. 10 multiply by p2 and substitute 2 sin 6 cos for sin 2 ; the equation then becomes p^ = 2 a2 p2 gin cos 0. 11. p = k cos e. 12. ^ = 3 tan-12. 13. p^ cos - = F. 1B2 ANALYTIC GEOMETRY [Ch. VI. EXAMPLES ON CHAPTER VI 1. Find the equation of the locus of 2xy-7x-\-iy = referred to parallel axes through the point (-2, |). 2. Transform the' equation x^ - 4xi/ + 4:if - Q x + I2y = to new rectangular axes making an angle tan-i | with the given axes. 3. Transform y'^ — xy — dx + by = i) to parallel axes through the point (-5, -5). Draw an appropriate figure. 4. Transform the equation of example 3 to axes bisecting the angles between the old axes. Trace the locus. 5. To what point must the origin be moved (the new axes being parallel to the old) in order that the new equation of the locus 2x2- 5a;^ _ 3^2_2a: + 13y - 12 = shall have no terms of first degree ? Solution. Let the new origin be (h, k) ; then x - x' + li, y = y' + k, and the new equation is 2(x' + hy-o(x'^h)i2j'+k)-3(y'^ky-2(x' + h) + ld(y'+k)-12 = 0, U., 2x'^ - 5x'y' - 3/2 + (^h - bk -2)x' - (bh + Qk- 13)/ + 2/i2 - 5M- - 3)^2 _ 2A + 13^ - 12 = 0; but it is required that the coefficients of x' and y' shall be ; i.e., h and k are to be determined so that ^h-bk- 2 = 0, and 5h-\-6k -13 = 0; hence A = -V- and k = f. Therefore the new origin must be at the point (^^, f), and the new equation is 2 x'2 - 5 x'y' - 3 /2 _ 8 = 0. 6. The new axes being parallel to the old, determine the new origin so that the new equation of the locus x'^ - 3 a-^ + ?/2 + 10 a: - 10 ?/ + 21 = shall have no terms of first degree. 7. Transform the equations x + y — 3 = and 2a; — 3?/ + 4 = to parallel axes having the point of intersection of these lines as origin. 8. Transform the equation j + ^ = 1 to new rectangular axes through the point (2, 3), and making the angle tan "^ ( — |) with the old axes. 9. Through what angle must the axes be turned that the new equa- tion of the line 6x + 4?/ — 24 = shall have no y-term ? Show this geometrically, from a figure. 76.] TBANSFORMATION OF COORDINATES 133 10. Through what angle must the axes be turned in order that the new equation of the line 6 a: + 4 y = 24 shall have no x-term ? Show analytically (cf. also examples 8 and 9). Solution. Let 6 be the required angle ; then the equations of trans- formation are X = x' cos 6 — y' sin 6 and y = x' sin + y' cos ; and the new equation is (6 cos ^ + 4 sin 0)x - (tj sin 6 - 4:cos0)y = 24: ; but it is required that the coefficient of x be 0, 6cos^ + 4sin ^ = 0, i.e., tan^ = — f; whence ^ = tan-i( — f), and the equation becomes (6 sin ^ - 4 cos ^) ^ + 24 = 0, which reduces to — ^:y + 24 = 0, V13 i.e., to 5z/ + 12Vl3 = 0. 11. Through what angle must the axes be turned to remove the a:-term from the equation of the locus Ax -\- By +C = 0"^ to remove the y-term ? 12. Show that to remove the xy-term. from the equation of the locus, 2 x^ — o xy — S y^ = S (cf. Ex. 5), the axes must be turned through the angle 6 = 67° 30', i.e., so that tan 2^ = — 1. What is the new equation? 13. Through v\^hat angle must a pair of rectangular axes be turned that the new x-axis may pass through the point ( — 2, — 5) ? 14. What point must be the new origin, the direction of axes being unchanged, in order that the new equation of the line Ax -{■ By -f C = shall have no constant term? 15. To what point, as origin of a pair of parallel axes, must a trans- formation of axes be made in order that the new equation of the locus, xy — y'^ — X -\- y = 0, shall have no terms of first degree? Construct the locus. 16. Find the new origin, the direction of axes remaining unchanged, so that the equation of the locus, x'^ + xy — 'd x — y -\- 2 = 0, shall have no constant term. Construct the figure. 17. Transform the equation 4^2 -f 2V3xy + 2y^ = 1 to new rectan- gular axes making an angle of 30° with the given axes, — origin unchanged. 134 ANALYTIC GEOMETRY [Ch. VL 76. 18. Transform y^ = Sx to new rectangular axes having the point (18, 12) as origin, and making an angle cot-i3 with the old. 19. Transform to rectangular coordinates, the pole and initial line being coincident with the origin and x-axis, respectively : (a) p^ = a^ cos 2 6, (/3) p2cos2(9 = a2, (y) p = ^-sin2^. Transform to polar coordinates, the ar-axis and initial line being coin- cident : 20. (x^ + y^y = k^(x^ - ?/2), the pole being at the point (0, 0) ; 21. a;2 +2/2 — 7 ^^^ pole being at the point (0, 0) ; 22. x^ + y^ = IQx, the pole being at the point (8, 0). 23. Transform the equation y^ + 4:ay cot 30° — 4 ax = to an oblique system of coordinates, with the same origin and a:-axis, but the new y-axis at an angle of 30° with the old x-axis. 24. Transform the equation t^ + ^ = 1> to new axes, making the positive angles tan -i | and tan ~^(— f), respectively, with the old a;-axis, the origin being unchanged. 25. Transform the equation 3 x2 + 10V3 xy - 7 2/2 = (18 - SOVS) x + (42 + 30\/3) ?/ + (42 + 90\/3) to the new origin (3, —3), with new axes making an angle of 30° with the old. 26. Transform the equation 3 a;^ + 8 x?/ — 3 2/^ = to the two straight lines which it represents, as new axes. 27. Transform ^ = 1 to the straight lines — = 0, as new 25 9 ^ 25 9 axes. 28. Transform to polar coordinates the equation y^ (2 a — x) = x^. 29. Transform to rectangular coordinates the equation p i=a(cos2^+ sin2(9). 30. Prove the formula for the distance in polar coordinates [1] by transformation of the corresponding formula [2] in rectangular coordi- nates. 31. Transform the equation x cos a + ?/ sin a= p io polar coordinates. CHAPTER VII THE CIRCLE « Special Equation of the Second Degree 77. It must be kept clearly in mind that one of the chief aims of an elementary course in Analytic Geometry is to teach a new method for the study of geometric properties of curves and surfaces. Power and facility in the use of such a new method are best acquired by applying it first to those loci whose properties are already best understood. Accord- ingly, the straight line having already been studied in Chapter V, the circle will next be examined. It will appear later that the circle is only a special case of the conic sections already referred to in Art. 48, and might, therefore, be advantageously studied after the general prop- erties of those curves had been examined ; the present order is adopted, however, because the student is already familiar with the chief properties of the circle. In solving the exercises of this chapter the student should use the analytic methods, even when purely geometric methods might sufhce, — he is learning to use a new instrument of investigation, and is not merely studying the properties of the circle. 78. The circle: its definition, and equation. The circle may be defined as the path traced by a point which moves in such a way as to be always at a constant distance from a given fixed point. This fixed point is the center, and the constant distance is the radius, of the circle. 135 136 ANALYTIC GEOMETRY [Ch. VII. To derive the equation from this definition, let (7= (A, k') be the center, r the radius, and JP = (a;, z/) any point on the ->^ curve. Also draw the ordinates ^ M^O and MP, and the line OE i I parallel to the a;-axis ; then I / OP = r ; [geometric equation] >/ but (Art. 26), M-, — M^ OP = V(^x - hy + (z/ - ky, hence V(a; — A)^ + (^ ~ ■^)^ = ^' 5 i.e., (x-7i)^ + (y-k)^ = r^,* . . . [31] which is the equation of the circle whose radius is r, and whose center has the coordinates h and k. With given fixed axes, equation [31] may, by rightly choosing A, k^ and 7% represent any circle Avhatever ; it is, therefore, called the general equation of the circle. Of its special forms one is, because of its very frequent applica- tion, particularly important ; this form is the one for which the center coincides with the origin : in that case h = k = 0, and equation [31] becomes ^2 + 2/2 = ^,2.1 . . . [32] * Equation [31] may be written in the form (X - hy + iy- ky - r2 = ; the first member then becomes positive if the coordinates of any point outside of the circle are substituted for x and y, it becomes negative for a point inside of the circle, and zero for a point on the circle. Hence the circle may be regarded as the boundary which separates that part of the plane for which the function (x — h)'^ + (^ — A;)^ — r^ is positive from the part for which this function is negative. The inside of the circle may therefore be called its nega- tive side, while the outside is its jiositive side (cf. foot-note, Art. 4.3). t If one is unrestricted in his choice of axes, then an equation of the form of [32] may represent any circle whatever, — the axes need merely be chosen perpendicular to each other and through its center; — equation [31] is more general in that, the rectangular axes being determined by other considera- tions, it may still represent any circle whatever. 78-79.] THE CIRCLE 137 EXERCISES First construct the circle, then find its equation, being given 1. the center (5, -3), the radius 4; 2. the center (0, 2), the radius |; 3. the center (3, -3), the radius 3 ; 4. the center (0, 0), the radius 5; 5. the center (-4, 0), the radius 1. 6. How are circles related for which h and k are the same, while ?- is different for each ? for which h and r are the same, while k differs for each ? 7. What form does the equation of the circle assume when the center is on the a:-axis and the origin on the circumference? when the circle touches each axis and has its center in quadrant 11? 79. In rectangular coordinates every equation of the form cc -+ 2/'^ + 2 Goc + 2 i^'i/ + C = represents a circle. The equa- tions of the circles already obtained (equations [31] and [32], as well as the answers to examples 1 to 5 and 7) are all of the form x^^-y'^ + 2ax+2Fy-{-C=0; . . . (1) it will now be shown that, for all values of ^, F^ and (7, for which the locus of equation (1) is real, this equation represents a circle. To prove this it is only necessary to complete the square in the ^-terms and in the ?/-terms, by adding G^ -f F^ to each member of equation (1), and then transpose C to the second member. Equation (1) may then be written in the form (x + G-y + (?/ + Ff = a'^ + F^- = (V(/2 + i^2_ (7)2 . , (2) which is (cf. equation [31]) the equation of a circle Avhose center is the point (— (r^, — ^), and whose radius is 138 ANALYTIC GEOMETRY [Ch. VII. Note 1. This circle is real only ii G^ + F^ - C>0; for, if G^+F-2- C<0, its square root is imaginary, and no real values of x and ,?/can then satisfy __ equation (2) ; while if 6'"^ + F^ - C = 0, then equation (2) reduces to (x+Gy + (y + Fy = 0, (3) which may be called the equation of a " point circle," since the coordi- nates of but one real point, viz. {-G, -F), will satisfy equation (3). If, however, G'^ + F'^ -C>0, then equation (1) represents a real circle for all values of G, F, and C, subject to this single limitation. Note 2. Every equation of the form Ax'^ -i-Ay^ + 2Gx -\- 2 Fij + C = Q represents a circle, for, by Art. 38, this equation has the same locus as C F G has a:^+ y^ -\- 2— x -\- 2 — y -^ — = 0, and this last equation is of the A A A form of equation (1). Hence, interpreted in rectangular coordinates, every equation of the second degree from which the term hi xy is absent, and in tvhich the coefficient of x^ equals that of /, represents a circle. 80. Equation of a circle through three given points. By means of equation [31], or of the equation ^2 + ^2 + 2fe + 2^^/+ 6^=0, . . . (1) which has been shown in Art. 79 to be its equivalent, the problem of finding the equation of a circle which shall pass through any three given points not lying on a straight line can be solved ; i.e., the constants h, k, and r, or (r, F, and Q, may be so determined that the circle shall pass through the three given points. To illustrate : lot the given points be (1, 1), (2, "l), and (3, 2), and let x^ + y"^ ^- "IGx + 2Fy ■\- C = Q be the equa- tion of the circle that passes through these points ; to find the values of the constants Gr, F, and C. Since the point (1, 1) is on this circle, therefore (cf. Art. 35), 79-80.] - THE CIRCLE 139 similarly, 4 + l+4(7-2i' + (7 = 0, and 9 + 4 + 6(74-41^ + (7 = 0. These equations give: 6^ = — |-, F ~ —^, and (7=4. Substituting these values, the equation of the required circle becomes its center is at the point (f, ^), while its radius is ^VTO- Note. The fact that the most general equation of the circle contains three parameters (Ji, k, and r, or G, F, and C, above) corresponds to the property that a circle is uniquely determined by three of its points. EXERCISES Find the radii, and the coordinates of the centers, of the following circles ; also, draw the circles. 1. x2 + 2/2 - 4 a: - 8 2/ - 41 = 0. ^. 2 (x^- + i/) = 7 y. 2. 3 x'^ + 3 y^ — 5 X —7 y -\- 1 = 0. 5. ax^ + ay'^ = bx -\- cy. 3. x^ + y^ = 3 (x + 3). 6. (x + yy + (x - y)'^ = 8 a^. 7. What loci are represented by the equations (x - hy + {y - ky = 0, and x2 + ?/2-2a; + 6?/ + 38 = 0? Find the equation of the circle through the points : 8. (1,2), (3, -4), and (5,-6); 9. (0, 0), (a, b), and (b, a) ; 10. (-6, -1), (0, 1), and (1, 0) ; 11. (10, 2), (3, 3), and having the radius 2. 12. Find the equation of the circle which has the line joining the points (3, 4) and ("1, 2) for a diameter. 13. Find the equation of the circle which touches each axis, and passes through the point (~2, 3). 14. A circle has its center on the line 3x -\- iy = 7, and touches the two lines x + y = 3 and x — y = 3] find its equation, radius, and center ; also draw the circle. 140 ANALYTIC GEOMETRY [Ch. VII. SECANTS, TANGENTS, AND NORMALS 81. Definitions of secants, tangents, and normals. A straight line will, in general, intersect any given curve in two or more distinct points ; it is then called a secant line to the curve. Let P^ and Pg ^6 two successive points of intersection of a secant line P^P^Q Fig. 64. with a given curve LP^P^ ... K; if this secant line be rotated about the point P^ so that P^ approaches Pj along the curve, the limiting position P^T which the secant approaches, as P^ approaches coincidence with P^, is called a tangent to the curve at that point. This conception of the tangent leads to a method, of extensive ap^Dlication, for deriving its equation, — the so- called " secant method." * Since the points of intersection of a line and a curve are found (Art. 39) b}^ considering their equations as simulta- neous, and solving for x and ^, it follows that, if the line is tangent to the curve, the abscissas of two points of intersec- tion, as well as their ordinates, are equal. Therefore, if the line is a tangent, the equation obtained by eliminating x or y between the equation of the line and that of the curve must have a pair of equal roots. If the given curve is of the second degree, then the equa- tion resulting from this elimination is of the second degree, and the test for equal roots is well known (Art. 9) ; but if the given equation is of a degree higher than the second, other methods must in general be used. A straight line drawn perpendicular to a tangent and * For illustration, see Art. 84. 81-82.] THE CIRCLE 141 through the point of tangency is called a normal line to the curve at that point. Thus, in Fig. 64, PiP^) ^i^z ^^^ se- cants, PjT is a tangent, and P^N d normal to the curve at P^ 82. Tangents : Illustrative examples. (1) To find the equation of that tangent to the circle x^ -\- y^ = b which makes an angle of 45° with the a:-axis. Since this line makes an angle of 45° with the a:-axis its equation is y = x -\- h, where h is to be determined so that this line shall touch the circle. Clearly, from the figure, there are two values of h (OB^ and OB2) for which this line will be tangent to the circle. According to Art. 81, these values of b are those which make the two points of intersection of the line and the circle become coincident. Considering the equations x^ + ^2 _ 5 and y = X + b simultaneous, and elimi- nating y, the resulting equation in x is x^+(x + by = o, i.e., 2x^ + 2bx-\-b^-5 = 0. The roots of this equation will become equal, i.e., the abscissas of the points of intersection will become equal (Art. 9), if 62 - 2 (62 _ 5) = 0, i.e., iib = ± VlO. The equations of the two required tangent lines are, therefore, y = x + VlO, and y = x—VlO. (2) To find the equations of those tangents to the circle x^ + y^ = Q y that are parallel to the line x+2y + ll=0. The equation of a line parallel tox + 2y + ll = 0isx + 2y-]-k = 0, where k is an arbitrary constant (Art. 62), and this line will become tangent to the circle, if the value of the constant k be so chosen that the two points in which the line meets the circle shall become coincident. Considering the equations x^ -\- y'^ = Qy and x-\-2y-\-k = simulta- neous, and eliminating x, the resulting equation in y is {- k -2yy + y^=Qy, i.e., 5 y^ + (4: k - Q) y + k^ = 0. The two values of y will become equal if (Art. 9) (4:k - 6)2-20 ^•2 = 0, i.e., if t^ + 15 y^ - 9 = 0, i.e., if k = — 6 ±3 Vo, and the two required tangent lines are : X -\-2y -e + 3V5 =0, and x + 2y - 6 - SVd = 0. 142 ANALYTIC GEOMETRY [Ch. VII. EXERCISES Find the equations of the tangents : 1. to the circle x'^ -f ?/2 _ 4^ parallel to the line a: + 2y4-3 = 0; 2. to the circle 3(x2 + 2/"-^) =^y<, perpendicular to the line x' + ?/ = 7; 3. to the circle x^ + y'^ -^ \0 x — Q y — 2 = {), parallel to the line y = '2x-1; 4. to the circle x^ + ?/2 — ^2^ ^j^d forming with the axes a triangle whose area is r^. 5. Show that the line y = x -\- cV2 is, for all values of c, tangent to circle x'^ + ?/2 = c^; and find, in terms of c, the point of contact. 6. Prove that the circle x'^ + y^-\-2x-\-2y-\-l = {) touches both coordinate axes ; and find the points of contact. 7. For what values of c will the line 3x — 4y + c = touch the circle x2 + ?/2 _ 8.r + 12?/ - 44 = 0? 8. For what value of r will the cu'cle x'^ -\- y^ = r^ touch the line J/ = 3 X — 5 ? 9. Prove that the line ax = b (y — b) touches the circle x (x — a) + y (y — b) = ; and find the point of contact. 10. Three tangents are drawn to the circle x'^ + y^ = 9] one of them is parallel to the x-axis, and together they form an equilateral triangle. Find their equations, and the area of the triangle. 83. Equation of tangent to the circle x^ + y^ = r^ in terms of its slope. The equation of the tangent to a given circle, in terms of its slope, is found in precisely the same way as that followed in solving (1) of Art. 82. Let m-^ be the given slope of the tangent, then the equation of the tangent is of the form 7/ = m^x + b, . . . (1) wherein J is a constant which must be so determined that line (1) shall intersect the circle 2;2 + i/2 = r2 . . . (2) in two coincident points. 82-83.] THE CIRCLE 143 Eliminating y between equations (1) and (2) gives x^ + (m^x + 5)2 = r^, ^.6., x^ (1 + m-^^) -\- 2 hm-^x + 5^ _ ^2 _ q . and the two values of x obtained from this equation will become equal (Art. 9) if Qm^hy - (1 + ^^2) (^2 _ ^2) ^ 0^ z.e., if 6 = ± r Vl + ^1^- Substituting this value of h in equation (1), it becomes y = miX ±r^/l + rni^* . . . [33] which is then, for all values of mi, tangent to the circle (2). This equation [33] enables one to write down immediately the equation of a tangent, of given slope, to a circle whose center^ is at the origin. E.g., to find the equation of the tangent whose slope ?nj = 1 = tan 45°, to the circle x^ + y^ = 5, it is only necessary to substitute 1 for m^ and Vo for r in equation [33]. This gives as the required equation ij = x± ViO [of. (1) Art. 82]. Note 1. If the center of the given circle is not at the origin, i.e., if its equation is of the form x^ + y^ + 2 Gx -{-2 Fy + C = 0, instead of x^+y^ = r% then the same reasoning as that employed above would lead to y + F = m^(x+G)±VG^ + F^-C-Vl-\-m^^ . . [34] as the equation of the required tangent. This equation might have been obtained also by first transforming the equation x"^ + y'^ + 2 Gx -\- 2 Fy -\- C = to parallel axes through the point (-G, -F) ; this would have given x'^ -i- y'^ = G^ + F'^ - C = r^ as the equation of the sayne circle, but now referred to axes through its center. Referred to these new axes y' = m^x' ±r Vl + m^^ (see eq. [33]) is, for all values of w^ tangent to this circle; transforming this last equation back to the original axes, i.e., substituting for x', y', and r their equals, viz., x -\- G, y -\- F, and V^r^ ^ p-i _ (j^ \^ becomes y + F=m^{x+ G) ± V^^ ^F'^-C . vTTw^ * This equation is sometimes spoken of as the magical equation of the tangent. 144 ANALYTIC GEOMETBT [Ch. VII. as before which is, for all values of 7n^, tangent to the circle whose center is at the point (-G, "F) and whose radius is \/G'^ + F'^- C. Note 2. Because of its frequent occurrence, it is useful to memorize equation [33]- On the other hand, it is not recommended that equation [34] be memorized, but it should be carefully worked out by the student. Instead of employing either of these formulas, however, the student may always attack the problems directly, as was done in Art. 82, EXERCISES Find the equations of the lines which are tangent : 1. to the circle x^ + y- = 16, and whose slope is 3 ; 2. to the circle x- + y- = 4, and which are parallel to the line x + 2 y + 3=0 (cf. Ex. 1, Art. 82); 3. to the circle x^ -{- y^ = 9, and which make an angle of 60*^ with the a;-axis ; with the ?/-axis ; 4. to the circle x'^ + y'^ = 25, and which are perpendicular to the line joining the points ("3, 7) and (7, 5) ; 5. to the circle x^ -^ y- = 2 x -\- 2 y — 1, and whose slojDe is ~1. 84. Equation of tangent to the circle in terms of the coordi- nates of the point of contact : the secant method. (a) Center of the circle at the origin. Let jP^ = (i^i, i/i) be the point of tangency, on the given circle x^ + y^ = r^. . . . (1) Through Pi draw a secant line LM, and let Pg = (2:2, ^2) be its other point of intersection with the circle. If the point -P2 raoves along the circle until it comes into coincidence with _Pi, the limiting position of the secant LM is the tangent PiT. (Art. 81.) The equation of the line L3I is y-y^^'hUVl^X-Xi). ... (2) 2^2 — .^1 Fig. 66. If now P2 approaches Pj until 83-84.] THE CIRCLE 145 X2=Xi and y2=yn equation (2) takes the indeterminate form y-yi=-^(p^-^i)' • • • (3) This indeterminateness arises because account has not yet been taken of the path (or direction) by which P2 shall approach Pi, and it disappears immediately if the condition that P2 is to approach Pi along the circle (1) is introduced. Since the fixed point P^ is on the circle (1), therefore ^i' + ^i'=^'; ... (4) and since P25 while approaching P^, always remains on circle (1), therefore x} ^ y^ = T^\ . , . (5) hence, subtracting equation (4) from equation (5), ^2 - ^i + y2 - Vi = 0, that is, (?/2 - y{) (^2 + yO = - (^2 - ^1) (^2 + ^1) ; whence, ^ — ^ = ^ ^^ . ^2-^1 ^2 + ^1 Substituting this result in equation (2) gives y-y^^-'^Q.-X,^,^ ... (6) which is the equation of the secant line LM of the given circle (1). * The difference between equations (2) and (6) consists in this : no mat- ter where the points (xi, yi) and (X2, 2/2) niay be, equation (2) represents the straiglit line passing through them ; but equation (6) is the equation of the line through these points only when they are on the circle orP- + y'^ = r^. In other words, equation (2) is the equation of the line passing through any two points whatever, while equation (6) is the equation of the line passing through any two points on the circumference of the circle. TAN. AN. GEOM. — 10 146 ANALYTIC GEOMETRY [Ch. VIL Now let P2 mo^'e along the circle until it coincides with Pi, i.e., until 2^2 = ^1' ^^^^ ^2 = ^/11 then equation (6) becomes i.e., y-yi= - —(x-x{), yi which, by clearing of fractions and transposing, may be written in the form xi^+yiy=^i-^yi^ i.e., i»ia? + 2/12/ = r2, . . . [35] which is the required equation of the tangent to the circle a;2 _|_ ^2 _ y.2^ ^^ ^^^ y^ being the coordinates of the point of tangency. (y8) Center of circle not at origin. If the equation of the given circle be x^ + i/+2ax + 2Fy + 0=^, ... (7) then, Pj and P^ being on this circle, ^^2 + ^^2+26^a:i + 2P^i + C=0, ... (8) and x^^+y^^ + ^ax^^ + 'lFy^ + C^O. . . . (9) Subtracting equation (8) from equation (9), x,^ - x{- + 2 (^(rr^ - 2:1) + y.^ - ^1^ + 2 P(^2 - ^1) = 0, which may be written in the form (^2 - ^i)(^2 + ^1 + 2 P) = - (a;2 - x{)(x^ + a^i 4- 2(7); whence, y^- y\ _ x^-\-x^-\-2G ^2-^1 2/2 + 2/1 + 2^ Substituting this result in equation (2) gives _x^±^h±l^r^_^^ . . . (10) ' ^2 + ^i + 2P^ 1^' ^ ^ 84-85.] THE CIRCLE 147 as the equation of the secant through the two points (x^^ y-^) and (a^g, ^2) ^^ '^^ circle (7). If, now, the point {x^^ y^ moves along the curve until it comes into coincidence with (j^v Vi)^ ^^^^ secant line becomes a tangent, and its equation is y-^i = - ^^_^^ (^-^i)- • • • (11) Clearing equation (11) of fractions, and transposing, it may be written thus : x^x + y^y + ax + Fy = x^ + y^ + ax^ + Fy^: . . . (12) but, by equation (8), the second member of equation (12) equals — G-x^ — Fy^ — C. Putting this value for the second member in equation (12), and transposing, that equation becomes xioc^-viv +G(dc + Xi)+F{y + yi) + C = 0, . . . [36] which is the required equation of the tangent to the circle (7), x-^ and y^ being the coordinates of the point of contact.* XoTE. Equation [36] may be easily remembered if it be observed that it differs from the equation of the circle [equation (7)] only in having x^x, y-^ij, x + x-^, and y + y^io. place of x^, y^, 2 a;, and 2y, respec- tively. Tt will be found later that any equation of the second degree (from which the a;y-term is absent) bears this same relation to the equa- tion of a tangent to its locus, x^ and y^ being the coordinates of the point of contact. Compare, also, equation [35] with equation (1). It must also be carefully kept in mind that equations [35] and [36] represent tangents only if (a'^, ?/j) is a point on the circle. It will be seen later that these equations represent other lines if (a-^, y^) is not on the circle. 85. Equation of a normal to a given circle. By definition (Art. 81) the normal at a given point, P-^ = (x-^^ y^), on any * Equations (11) and (12) are, of course, but different forms of the equa- tion of the same tangent as that represented by equation [36]. 148 ANALYTIC GEOMETRY [Ch. VII. curve is the line through P^, and perpendicular to the tangent at P^. Hence, to get the equation of the normal at any given point, it is only necessary to write the equation of the tangent at this point (Art. 84), and then the equa- tion of a line perpendicular to this tangent (Arts. 53, 62) and passing through the given point. Thus the equation of the normal to the circle x^ + f + 2ax+2Fi/ + O=0, . . . (1) at the point P^ =(ix-^, «/i), is The coordinates — Gr and — P of the center of the given circle (1) satisfy equation (2) ; hence^ every normal to a circle jmsses through the center of the circle. If the center of the circle be at the origin, then (7 = 0, P = 0, and Q=—r'^, and the equation (2) of the normal becomes which reduces to x^y — xy^ = 0, — an equation which could have been derived for the circle x'^ + y'^ = r^ in precisely the same way that equation (2) was derived from equation (1). EXERCISES 1. Derive, by the secant method, the equation of the tangent to the circle x^ -}- y^ = 2 rx, the point of contact being P^ = (x^, y^ . 2. Write the equation of the tangent to the circle : (a) x^ + ?/2 = 25, the point of contact being (3, 4) ; (^) a;2 + ?/2 - 3 a; + 10 ?/ = 15, the point of contact being (4, -11) ; (y) (x - 2)2 +(y - 3)2 = 10, the point of contact being (5, 4) ; (8) 3 a:2 + 3 2/2 - 2 2/ - 4 a: = 0, the point of contact being (0, 0). 85-86.] THE CIRCLE 149 3. Find the equation of the normal to each of the cu'cles of Ex. 2, through the given point. 4. A tangent is perpendicular to the radius drawn to its point of contact. By means of this fact, derive the equation of the tangent to the circle {x — ay^{y — by" = r'^ at the point {x^,y^) (cf. equation [36]). 5. From the fact that a normal to a circle passes through its center, find the equation of the normal to the circle x^ + y^ — 6 a; + 8^ + 21 =0 at the point (1, "4). 6. Find the equations of the two tangents, drawn through the ex- ternal point (11, 3) to the circle a;^ ■\- if- — 40. Suggestion. Use the equation of the tangent in terms of its slope. 7. What is the equation of the circle whose center is at the point (5, 3), and which touches the line 3a; + 2?/ — 10 = 0? X v 8. Under what condition will the line - + ;- = ! touch the circle a a;2 + ?/2 = 7'2 9 9. Find the equation of a circle inscribed in the triangle whose sides are the lines x = 0, y = 0, and _ _(- ^ = 1 . a b 10. Solve Ex. 6 by assuming x-^^ and y-^ as the coordinates of the point of contact, and then finding their numerical values from the two equa- tions which they satisfy. 86. Lengths of tangents and normals. Subtangents and subnormals. The tangent and normal lines of any curve extend indefinitely" in both directions ; it is, however, convenient to consider as the length of the tangent the length TP^, measured from the point of intersection (T) of the tangent with the x- axis to the point of tangency (Pj), and similarly to consider as the length of the normal the length P^N^ measured from P^ to the point of intersec- tion (iV) of the normal with the 2;-axis. 150 AJSTALYTIC GEOMETRY [Ch. VII. The subtangent is tlie length TM, where M is the foot of the ordinate of the point of tangency P^ ; and the subnormal is the corresponding length MN. As thus taken, the sub- tangent and the subnormal are of the same sign ; ordinarily, however, one is concerned merely with their absolute values, irrespective of the algebraic sign. The subtangent is the projection of the tangent length on the 2:-axis, and the sub- normal is the like projection of the normal length. 87. Tangent and normal lengths, subtangent and subnor- mal, for the circle. The definitions given in the preceding article furnish a direct method for finding the tangent and normal lengths, as well as the subtangent and subnormal, for a circle. ^•^., to find these values for the circle x^-\- 1/^ = 25, and correspond- ing to the point of contact (3, 4), proceed thus: The equation of the t^ni- gentP^Tis (Art. 84) Fig. 68. hence the a:-intercept of this tangent, ^.e., OT, = ^^- ; therefore the subtangent TM, which equals OM — OT^ is 3 — ^, ^.e., — 5J. The tangent length TP^ = ^MT^ + MPl'' = V(J^)2 + 42 = 6|. To find the normal length, and the subnormal, first write the equation of the normal at the point (3, 4); it is (Art. 86) 4:x — Sy = 0. Hence its a;-intercept is zero, and the subnormal, MO in this case, is — 3 ; the normal length P^ is 5. 86-88] THE CIRCLE 151 Similarly, corresponding to the point (x^, y-^ on the circle a;2 _|. ^2 __ ^2^ ^]^Q subtangent = — ^, the tangent length X. ^^] = -^, the subnormal = — x^^ and the normal length = r. The derivation of these values is left as an exercise for the student, as is also the derivation of the corresponding expressions for the circle x^ -\- y^ -\-1 Gx + 2 Fy + C^ = 0, the point of contact being {x^^ y^. EXERCISES Find the lengths of the tangent, subtangent, normal, and subnormal, 1. for the point (4, -11) on the circle a:^ + ?/2 — 3 a; + 10 ?/ = 15 ; 2. for the point (1, 3) on the circle x'^ + y'^ — V) x =0] 3. for the point whose abscissa is V7 on the circle a;- + y^ = 25. 4. The subtangent for a certain point on a circle, whose center is at the origin, is 5i, and its subnormal is 3. Find the equation of the circle, and the point of tangency, 88. To find the length of a tangent from a given external point to a given circle. Let P-^ = {x-^, y^ be the given external point, and let a;2 + y2 _|_ 2 fe + 2 ^^ + C = be the given circle. The center of this circle (Art. 79) is (~(r, ~-F), and its radius is center K^ draw the tangent Pj§, and also the radius KQ. Then P^ = KP^ - KQ^ ; but (Art. 26) and o Fig. 69. ^= (72-hP2_^. (Art. 79) 152 ANALYTIC GEOMETBT [Ch. VII. i.e., the square of the length of the tangent from a given external point to the circle x^ + 7/^4-2 Gix + 2 Fy + C = * is obtained by writing the first member only of this equation.^ and substituting in it the coordinates of the given point. f 89. From any point outside of a circle two tangents to the circle can be drawn, (a) Let the equation of the circle be 0:2 + ^2^^2, . . . (1) then (Art. 83) the line y = mx + r Vl + ni^ . . . (2) is, for all values of m, tangent to this circle. Let P^={x^, ?/j) be any given point outside the circle (1); then the tangent (2) will pass through P^ if, and only if, m be given a value such that the equation y^ = mxi + rVl -f 711^ . . . (3) shall be satisfied. Transposing, squaring, and rearranging equation (3), it is clear that it will be satisfied if, and only if, m is given a value such that the equation (r^ — x-^^m^ + 2 x-^y^m + r^ — y^ — is satisfied; i.e., equation (3) is satisfied if, and only if. ^ ^ - ^^y^ ± ^v^i + Va - ^'\ ... (4) — x^y-^ ± r^x^ + yf — r^ r^ — x^ Equation (4) gives two, and only two, real values for m when {xy^ y^ is outside of the circle, for then x^ + y^ — r^ is * If the circle is given by the equation Ao^ + Aip- + 2 G^x -f 2 F?/ + C = 0, it must first be divided by A before applying this theorem. t The expression x^ + ?/i^ + 2 Gx\ + 2 Fy\ + (7 is called the power of the point Pi= (xi, yi) with regard to the circle x'^ + y"^ -\- 2. Gx -\- '2. Fy -{- (7 = 0. 88-89.] THE CIRCLE 153 positive (Art. 78, foot-note) ; these values of m, being sub- stituted in turn in equation (2), give the two tangents through Pj to the circle (1). If Pj is 071 the circle (1), then x-^ -{- 7/^ — r^ = 0; hence the two values of m from equation (4) coincide, and the two tangents also coincide, i.e., there is in this case but 07ie tangent. If P^ is within the circle, then the two values of m from equation (4) are both imaginary and no tangent through Pj can be drawn to the circle (1).* If either value of m from equation (4) is substituted in equation (2), and then equations (2) and (1) are considered as simultaneous and solved for x and ^, the coordinates of the corresponding point of contact are obtained. Note. The properties of the equations of the Une and circle have thus established a geometric property of the circle [cf. Art. 31, (III)]. (^) If the equation of the given circle had been x^ + f-\-2ax + 2F^+O = 0, . . . (5) it could, by Art. 71, have been transformed to new axes through its center (~(7, ~P) and parallel respectively to the given axes ; its equation would thus have become X 12 I ^,/2 _ J2, + y2 = r2, . . , (6) where x^ and y^ refer to the new axes. This transformation, however, leaves the circle and all its intrinsic properties unchanged ; but (a) applies to circle (6), hence it is proved that circle (5), which is circle (6) merely referred to other axes, has the same properties. * These conclusions may also he stated thus : if Pi is outside of the circle, equation (4) gives two real and distinct values for m ; corresponding to these there are two real and distinct tangents ; if Pi is on the circle, the two values of m are real but coincident, and there are two real but coincident tangents ; if Pi is inside of the circle, the two values of m are imaginary, and the two corresponding tangents are therefore also imaginary. 154 ANALYTIC GEOMETRY [Ch. VIL 90. Chord of contact. If two tangents are drawn from any external point to a circle, the line joining the two corre- sponding points of tangency is called the chord of contact for the point from which the tangents are drawn. The equation of this chord of contact may be found by first finding the points of tan- gency and then writing the equation of the straight line through those two points. It may, however, be found more briefl}^, and much more ele- gantly, as follows : Let P^ = (tj, y^) be the given external point from which the two tangents are drawn ; and let T^ = (x^^ y^ and T^ = (.rg, ^3) be the points of tangency on the circle x^ + f + 2ax-\-2Fy-\-C=0; . . . (1) it is required to find the equation of the line passing through T2 and 2^3. The equation of the tangent at T^ is (Art. 84) ^2^ + I/2I/ + ^(^ + ^2) + ^(1/ +1/2) + C^ = 0, . . . (2) and the equation of the tangent at T^ is But each of these tangents passes through the point Pj ; hence its coordinates, x^ and ^j, satisfy equations (2) and (3), therefore ^1^2 + ^1^2 + ^(^1 + ^2) + ^(1/1 + ?/2) + ^ = 0, . . . (4) and x^x^ + y^y^ -f G-Qx^ + x^} -f F(y^ + 2/^^-\- C =0. . . . (5) Equations (4) and (5), however, assert respectively that (^2' ^2) ^^^ C^3^ ^s) ^^^ points on the locus of the equation ^i^ + ^i^ + ^(^i + ^) + ^(^i + «/)+C = 0. . . . (6) 90.] THE CIRCLE 155 But equation (6) is of the first degree in the two varia- bles X and ?/, hence (Art. 57) its locus is a straight line, and, since it passes through both T-^^ = (x2'> I/2) ^^^ ^3=(^3' ^3)' it is the equation of the chord of contact ; i.e., x^x + y^y + G-{x + x^) -^F{y + y^) + C = . . . [37] is the equation of the chord of contact corresponding to the external point P^^(x^^ y^. It is to be noticed that if Pj is on the circle, then the two tangents drawn through it coincide with each other and with the chord of contact ; the equation of the chord of con- tact [37] then becomes the equation of the tangent at P^, as it should (cf. equation [36]). If, then, (a^j, yj) is a point on the circle (1), equation [37] is the equation of the tangent to the circle at that point ; if, on the other hand, (x^, y^ is outside of this circle, then equation [37] is not the equation of a tangent, but of the chord of contact corresponding to that external point. EXERCISES 1. Find the length of the tangent from the point (8, 10) to the circles : 2. (a) Write the equation of the chord of contact corresponding to the point (5, 6) for the circle x'^ + y'^ — Qx — 4.y = 8. (/3) Find the coordinates of the points in which this chord cuts the circle. (y) Write the equations of the tangents to the circle at these points of intersection ; show that these lines pass through the given point (5, 6). 3. By the method of exercise 2, find the equations of the tangents drawn to the circle (o x — 2)^ + (3 ?/ + 5)2 — 4^ from the origin ; from the point (1, 2). 4. Find the locus of a point from which the tangents drawn to the two circles 2 ^2 + 2 ?/2 - 10 X + 14 y + 35 = and x2 + ?/2 = 9 are of equal length. Show that this locus is a straight line perpendicular to the line joining the centers of the given circles. 156 ANALYTIC GEOMETRY [Ch. VII. 5. For what point is the line 3 a; + 4?/ = 7 the chord of contact with regard to the circle x- + ^^ = 14 ? 6. Find the chord of contact for the circle x^ + y- = 25, corresponding to the point (3, 7) ; to the point (3, 2). 7. By means of the equation y — y-^ = 7n(x— x-^) prove that two tan- gents can be drawn through the external point (.r^^, y^) to the circle whose equation is x'^ -\- y^ = r^- 8. Solve (/3) and (y), of exercise 2, by means of the equation y — Q = m(x — 5). 91. Poles and Polars. If through any given pomt Pj = (2:p^j), outside, inside, or on the circle, a secant is drawn, meeting the circle in two points, as Q and R^ and if tan- gents are drawn at Q and R^ they will intersect in some point as The locus of P', as the secant revolves about Pj, is called the polar of Pj with regard to the circle ; and Pj is the pole of that locns. It will be proved in the next article that the locus of P' is a straiglit line whose equation is of the same form as that of the tangent (Art. 84), and as that of the chord of contact (Art. 90) already found. 92. Equation of the polar. Let P^ = (x^, y{) be the given point, the equation of whose polar, with regard to the circle x^ + y'^ + 2ax^2Fy-{-C=0, . . . (1) is sought. Also let P^QR be any position of the secant through Pj, and let the tangents at Q and R intersect in P'=(a;', y); then the equation of P^QR (Art. 90) is x'x + y'y + a(x^x'^-{-F(iy + y')+C=^. ... (2) Fig. 71. 90-93.] THE CIRCLE 157 Since P^ is on this line, therefore ^i^' + yi/ + ^(^i + ^0 + -^(yi + yO + C^=0. . . . (3) Equation (3) asserts that the coordinates, x' and y\ of P' satisfy the equation . ^i^ + yi^ + ^(^ + ^i) + -^(y + yi)+C^=0; . . . [38] i.e.^ this variable point P' always lies on the locus of equa- tion [38] ; in other words, [38] is the equation of the polar of P-^ with regard to the circle (1). Moreover, since equation [38] is of the first degree in the variables x and y, therefore (Art. 57) its locus is a straight line ; that is, the polar of any given pointy with regard to any given circle^ is a straight line. That equations [36] and [37] have the same form as equa- tion [38] is due to the fact that the tangent and the chord of contact are only special cases of the polar. 93. Fundamental theorem. An important theorem con- cerning poles and jDolars is : If the polar of the point P^, with regard to a given circle^ passes through the point P^., then the polar of P^ passes through P^. Let the equa- tion of the given circle be x^ + y'- + 2ax + 2Fy -h(7=0, . . . (1) and let the two given points be Pi = Cx^, ^i), and P2 = (.^2^1/2)1 then (Art. 92) the equation of the polar of P^ is x^x+y^y + a(x-\-x^) + F(y-^y^)+C=0. ... (2) 158 ANALYTIC GEOMETRY [Ch. VIL If this line passes through P^, then But the equation of the j)olar of P^ (Art. 92) is x^x + y,j + a(x + x^^^-Fiy + y,J + O=0, ... (4) and equation (3) proves that the locus of equation (4) passes through Pj, which establishes the theorem. EXERCISES 1. Find the polar of the point (6, 8) with reference to the circle x^ + y^ = 14. 2. Find the x:>olar of the pohit (1, 2) with regard to the circle x^ + y^ -\- ^ X — Q y — 10. 3. Find the pole of the line 4x + 6?/ = 7, and of the line aa: + % — 1 = 0, with regard to the circle x'^ + ?/^ = 35. 4. Find the equations of the two tangents to the circle x^ + y^ = 65 from the point (4, 7); from the point (11, 3). 5. Show that if the polar of (Ji, k) with respect to the circle x'^+y'^ = c^ touch the circle 4 (x'^ + y^) = c^, then the pole (h, k) will lie on the circle ■x^ + y2 — 4(^2. 6. Show that the pole of the line joining (5, 7) and (~11, 1) is the point of intersection of the polars of those two points with reference to the circle x'^ + ?/- = 100. 7. Find the pole of the line 2x — 3y = with respect to the circle x^ + y^- = 9. 8. Show what specialization of a polar converts it into a chord of contact, and what further specialization converts it into a tangent. 94. Geometrical construction for the polar of a given point, and for the pole of a given line, with regard to a given circle. Since tlie relation between a polar and its pole (see def. Art. 91) is independent of the coordinate axes, therefore the given circle may, without loss of generality, be assumed to have its center at the origin. If P^ = (a?j, ^j) is any given point, and a^-i-f = r^ . . . (1) 93-94.] THE CIRCLE 159 is a given circle, whose center is at the point 0, then the equation of OP^ (Art. 51) is y^x — x^y^O. ... (2) /^ Y ^ / Xk / Py \ \ I X \ \ \ ^ ^^ ) \ ^ — Fig. 73.^ Fig. 73; Let LL^ be the polar of P^, with regard to the given circle, and let it meet OP^ in K. The equation of LL-^ (Art. 92) is x^x-\-y^y=-r^, . . . (3) Equations (2) and (3) show (Art. 62) that LL^ and OP^ are perpendicular to each other; ^.e., the line joinmg the given point P^ to the center of the circle is perpendicular to the polar of P^ with regard to the circle. The distance (OIC) from the origin to the line LL^ (Art. 04) is ■\/x^ 4- y-^ and the length of OP-^ (Art. 26) is Vx^+y} C^) (5) therefore OK' OP^^ ,,.2 ■\/x^ + y-^ = /■■ -Vx^^ + yi^ Hence, to construct, with regard to a given circle, the polar of any given point P^, join that point to the center of the circle, then on OP^ (produced if necessary) find a point K such that the rectangle OP^ • OK is equal to the square 160 ANALYTIC GEOMETRY [Ch. VII. on the radius of the circle, and through K draw a line perpendicular to OPj ; this line is the required polar. Similarly the pole may be constructed, if the polar and the circle are given. 95. Circles through the intersections of two given circles. Given two circles whose equations are x' ^ f -^ 2a,x -{- 2F,i/ ^ C, = 0, . . . (1) and x' + f-{-'2 a^x + 2F^y -\- 0^ = 0. , . . (2) These circles intersect, in general, in two finite j)oints P-^ = (xi, 7/i) and P2 = Qx2, ?/2), and (Art. 41) the equation x'^f^ + 2a,x+2F,y-\- C\ + k(x' + f^ + 2a2X^2F.^-i-CO = 0, ... (3) where k is any constant, represents a curve which passes through these same points Pi and P2. The locus of equation (3) is, moreover, a circle (Art. 79) ; hence, a series of different values being assigned to the param- eter ^, equation (3) represents what is called a "family" of circles ; each one of these circles passing through the two points Pi and P2 in which the given circles (1) and (2) intersect each other. 96. Common chord of two circles. If in equation (3), Art. 95, the parameter Jc be given the particular value — 1, the equation reduces to 2 ((7i - a2')x + 2(Fi- F2^y + (7i - a=0, . . . . (4) which is of the first degree, and therefore represents a straight line ; but this locus belongs to the family repre- sented by equation (3) of Art. 95, hence it passes through the two points Pi and P2 in which the circles (1) and (2) inter- sect. This line (4) is, therefore, the common chord* of these circles. 94-97.] THE CIRCLE 161 To obtain the equation of the common chord of two given circles it is, then, only necessary to eliminate the terms in x^ and y^ between their equations. E.g., to find the common chord of the circles 2a:2 + 2y2+ 3a:+ 5?/- 9 = 0, . . . (a) and 6x2 + 6 3/2 + 11 a: + 13 2/ -23 = 0, • • • (/?) multiply equation (a) by 3 and subtract the result from equation {^) ; this gives x-?/ + 2 = 0, . . . (y) as the equation of the common chord of the given circles. This result may be verified by finding the points of intersection (Art. 89) of the circles (a) and (/?), and then writing the equation of the straight line through those two points. Since the common chord of two circles intersects each of these circles in the points in which they intersect each other, therefore the points of intersection of two circles may be found by finding the points in which their common chord intersects either of them. E.g., to find the points in which the circles (a) and (y8) intersect each other, it is only necessary to find the points in which (y) cuts either (a) or (/3). 97. Radical axis ; radical center. The line whose equation is obtained by eliminating the x^ and y"^ terms between the equations of two given circles, as in Art. 96, whether the circles intersect in real points or not, is called the radical axis of the two circles. If the two given circles intersect each other in real points, then this line is also called their com- mon chord ; that is, the common chord of two circles is a special case of the radical axis of two circles. * Equation (3) of Art, 95, which for every value of k represents a circle passing through the two points in which the given circles (I) and (2) inter- sect, may be written in the form x2 + y2 + 2^L^tM.x + 2^1+l^^+^^hl^=0. 1 + ^- i + ic ^^ I +k The coordinates of the center of this circle are (Art. 79) \+k 1 +k If then k be made to approach —1, both of these coordinates approach infinity, but the circle always passes through the two fixed points in which the given circles intersect ; hence the common chord of two given circles may be regarded as an infinitely large circle whose center is at mfinity. TAN. AN. GEOM. 11 162 ANALYTIC GEOMETRY [Ch. VII. Three circles, taken two and two, have three radical axes. It is easily shown that these three radical axes pass through a common point ; this point is called the radical center of the three circles. EXERCISES 1. Find the equation of the common chord of the cu'cles 2. Find the point of intersection of the circles in exercise 1, and the length of their common chord. 3. Find the radical axis, and also the length of the common chord, for the circles x^ + y^ + ax -i- by -\- c = 0, x^ -i- y'^ -\- bx -\- ay + c = 0. 4. Find the radical center of the three circles a:^ + ?/2 + 4 a: + 7 = 0, 2 (a;2 + 2/2) + 3 2- + 5y + 9 = 0, x^ + y- + y — 0. 5. Show that tangents from the radical center, in exercise 4, to the three circles, respectively, are equal in length. 6. Prove analytical!}^ that the tangents to two circles from any point on their radical axis are equal. 7. Find the polar of the radical center of the circles in exercise 4, with respect to each circle. 8. Prove analytically that the three radical axes of three circles, the circles being taken in pairs, meet in a common point. 98. The equation of a circle : polar coordinates. Let OB be the initial line, the pole, C=(p^, 6{) the center of the circle, r its radius, and P = (p, ^) any point on the circle. Draw 0(7, OP, and OP ; then, by trigonometry, r^ =: p^ + p^^ — 2 pp^ cos (^ — ^j), i.e., /32_2/)i/9COS(6>- (9i) + ^^2 _ ^2 ^ 0, . . . [39] which is the equation of the given circle. Fig. 74. 97-99.] THE CIRCLE 163 Depending upon the relative positions of the polar axis, the pole, and the center of the circle, equation [39] has several special forms : (a) If the center is on the polar axis, then 6-^ = 0, and equation [39] becomes p^ — 2 p^p cos ^ + p^2 _ ^2 _ ; (/3) If the j)ole is on the circle, then p-^ = r, and equa- tion [39] becomes /3- 2rcos(6'-6'i)= 0; (7) If the pole is on the circle and the polar axis a diame- ter, then pi = r and 0^ = 0, and equation [39] becomes p —2r cos6 = ; (5) If the center is at the pole, then pi = and equation [391 becomes L J p = r. 99. Equation of a circle referred to oblique axes. Let the axes OX and Oy be inclined at an angle co ; let C = {h, k^ be the center of the circle, r _ its radius, and P = (ic, ^) any point on the circle. Draw the ordinates M^Q and MP^ connect C'and P, and draw CHL paral- lel to the a;-axis ; then Fig. 75. -\- 2 OR ' RP COS CO ; hence r^ = (x — h)"^ -\- Qy — k^ + 2(x — K){y — k) cos co, {.e.,(x-hy+(y-ky^-\-2<^x-h)(2j-k)Gosco-r^ = 0;, ..[40] which is the equation of the given circle. 164 ANALYTIC GEOMETRY [Ch. VII. It is to be observed that this equation [40] is not of the form x^^yij^2ax + 2Fy + (7= 0, which was discussed in Art. 79 ; it differs from that equa- tion in that it contains an a^y-term. If, however, the axes are rectangular, as in Art. 79, then cos o) = 0, and equation [40] reduces to the standard form of Art. 79, viz. : a;2 ^ ^2 _|_ 2 (7a; + 2 IV + (7= 0, which is a special case of equation [40]. 100. The angle formed by two intersecting curves. By the angle between two intersecting curves is meant the angle formed by the two tangents, one to each curve, drawn through the point of intersection. Hence to find the angle at which two curves intersect, it is only necessary to find the point of intersection, then to find the equations of the tangents at this point, one to each curve, and finally to find the angle formed by these tangents. EXERCISES 1. Find the polar equation of the cii'cle whose center is at the point ( 7, -T j and whose radius is 10 ; determine also the points of its inter- section with the initial line. 2. Find the polar equation of a circle whose center is at the point ( 15, - J and whose radius is 10. Find also the equations of the tangents to the circle from the pole. 3. A circle of radius 3 is tangent to the two radii vectores which make the angles 60° and 120° with the initial line : find its polar equa- tion, and the distance of the center from the origin. 4. Find the equation of a circle of radius 5, with center at the point (2, 3), if o) is 60°. 5. Find the equation of a circle of radius 2, with center at the origin, if (o is 120°. 99-100.] THE CIRCLE ' 1(J5 6. Determine the equation of tlie circle circumscribing an equilateral triangle, — the coordinate axes being two sides of the triangle. 7. A circle is inscribed in a square. What is its equation, if a side and adjacent diagonal of the square are chosen as the y- and ^--axis, respectively? What are the coordinates of the points of tangency? 8. Find the angle at which the circle x^ + 3/2 = 9 intersects the circle (x- — 4)- + y- — 2 1/ = 15. At what angle does the second of these circles meet the line x + 2 y = ^1 EXAMPLES ON CHAPTER VII 1. Find the equation of the cipcle circumscribing the triangle whose vertices are at the points (7, 23^"1) ~4), and (3, 3). What is its center? its radius ? 2. Determine the center of the circle (^x + a)2 + (?/ + &)2 = a2 + i\ What family of circles is represented by this equation, if a and h vary under the one restriction that a^ + &2 is to remain constant ? 3. What must be the relations among the coefficients in order that the circles x2 + 2/2 + 2Gia; + 2i^,?/+ C^ = 0, and ^2 + ?/2 + 2 GgX + 2 F^y + Cg = 0, shall be concentric ? that they shall have equal areas ? 4. Under what limitations upon the coefficients is the circle Ax"^ + Ay'^ + Dx^ Ey -V F=0 tangent to each of the axes ? 5. Find the equation of the circle which has its center on the x-axis, and which passes through the origin and also through the point (2, 3). 6. Find the points of intersection of the tw^o circles a;2 ^ ^2 _ 4 3, _ 2 ?/ _ 31 = and x2 + ?/2_4x + 2?/ + l=0. 7. Circles are drawn having th ir centers at the vertices of the triangle (7, 2), (-1, -4) and (3, 3), respectively, and each 23assing through the center of a fourth circle which circumscribes this triangle ; find their equations, their common chords, and their radical center. 8. Circles having the sides of the triangle (7, 2), (-1, -4), (3, 3) as diameters are drawn ; find their equations, their radical axes, and their radical center. 166 ANALYTIC GEOMETRY [Ch. VII. 9. Find the equation of the circle passing through the origin and the point {x-^, y-^), and having its center on the 2/-axis. 10. The point (3, "5) bisects a chord of the circle x'^ -\- ]f- — 277 ; find the equation of thab chord. 11. A circle touches the line 4x + 3?/ + 3 = at the point (~3, 3) and passes through the point (5, 9) ; find its equation. 12. A circle, whose center coincides with the origin, touches the line 7a; — ll?/ + 2 = 0; find its equation. 13. At the points in which the circle x'^ + y'^ — ax — hy = cuts the axes, tangents are drawn ; find the equations of these tangents. 14. A circle, whose radius is 7, touches the line Dy = lx — 1 at the point (8, 11) ; find the equation of this circle. 15. A circle is inscribed in the triangle (7, 2), (—1, "•!), (3, 3); find its equation ; find also the equations of the polars of the three vertices with regard to this circle. 16. Through a fixed point (Xy, y^) a secant line is drawn to the circle ^2 _|. ^2 _ ^2 . -^Yi^ the locus of the middle point of the chord which the circle cuts from this secant line, as the secant revolves about the given fixed point (x^, y-^) . 17. Prove analytically that an angle inscribed in a semicircle is a right angle. 18. Prove analytically that a radius drawn perpendicular to a chord of a circle bisects that chord. 19. Show that the distances of two points from the center of a circle are proportional to the distances of each from the polar of the other. 20. Two straight lines touch the circle x^ -\- y'^ — 5 x — 3 y -\- Q = 0, one at the point (1, 1) and the other at the point {2, 3) ; find the pole of the chord of contact of these tangents. 21. Find the condition among the coefficients that must be satisfied if the circles x'i + y'i + 2 GyX + 2 F^y =0 and x"^ + y'^ + 2 G.^x + 2F^y = Q shall touch each other at the origin. 22. Determine G, F, and C so that the circle x^- + y'^ + 2Gx-\-2Fy^ C = shall cut each of the circles a;2^^2_4a;_2y + 4 = and x^ -h y- + 4:X + 2 y = 1 at right angles (cf. Art. 100). 100.] THE CIRCLE 167 23. Given the two circles a:2 + ?/2-4x-2?/ + 4=0 and x^ ■}- y'^ -\- ^x +2 y - ^ = 0'^ find the equation of their common tangents. 24. Find the radical axis of the circles in example 23 ; show that it is perpendicular to the line joining the centers of the given circles, and find the ratio of the lengths of the segments into which the radical axis divides the line joining the centers. How is this ratio related to the radii of the circles ? Is this relation true for any pair of circles what- ever ? 25. Given the three circles : a;2 + / - 16 a: + 60 = 0, 3 x^ + 3 ?/2 _ 36 a; + 81 = 0, and a;2 + ?/2 _ 16 a; - 12 ?/ + 84 = ; find the point from which tangents drawn to these three circles are of equal length, also find that length. How is this point related in position to the radical center of the given circles ? Prove that this relation is the same for any three circles. 26. Find the locus of a point which moves so that the length of the tangent, drawn from it to a fixed circle, is in a constant ratio to the dis- tance of the moving point from a given fixed point. 27. Let P be a fixed point on a given circle, T a point moving along the circle, and Q the point of intersection of the tangent at T with the perpendicular upon it from P ; find the locus of Q. Suggestion. Use polar coordinates, P being the pole, and the diam- eter through P the initial line. 28. Find the length of the common chord of the two circles (x — rt)2 + (?/ — bY = T^ and {x — 6)^ + (^ — «)^ = ^^• From this find the condition that these circles shall touch each other. 29. If the axes are inclined at 60°, prove that the equation x'^ -\- xy -\- y"^ — ^:X —^ y — 2 — ^ represents a circle ; find its radius and center. 30. What is the obliquity of the axes if the equation a;2 + V3 a:2/-f?/"2 — 4a; — 63/-|-5 = represents a circle? AVhat is its radius? 31. For what point on the circle a:^ + ?/2 _ 9 ^^-e the subtangent and the subnormal of equal length? the tangent and normal? the tangent and subtangent ? 108 ANALTTJC GEOMETRY [Ch. VH. 32. An equilateral triangle is inscribed in the circle a:^ + ?/2 — 4 with its base parallel to the a:-axis ; through its vertices tangents to the circle are drawn, thus forming a circumscribed triangle ; find the equations, and the lengths, of the sides of each triangle. 33. The poles of the sides of each triangle in example 32 are the vertices of a triangle ; find the equations of its sides, and draw the figure. 34. A chord of the circle a;^ + ?/2 — 22 a: — 4 !/ + 25 = is of length 4 V5, and is parallel to the line 2x + ^ + 7 = 0; find the equation of the chord, and of the normals at its extremities. 35. Find the equation of a circle through the intersection of the circles ^2+ ?/2-4 = 0, x^-\-y'^ — '2.x— ^y+b = 0, and tangent to the line x + y-o = i). 36. The length of a tangent, from a moving point, to the circle a;2 + ^2=6 is always twice the length of the tangent from the same point to the circle a:- + 3/^ + 3 (a: + ?/) = 0. Find the equation of the locus of the moving point. 37. Find the locus of the vertex of a triangle having given the base = 2 «, and the sum of the squares of its sides = 2 J^. 38. Find the locus of the middle points of chords drawn through a fixed point on the circle x^ + 3/'^ = a^. 39. Through the external point P^ = (xj, y^), a line is drawn meeting the circle x'^ + y'^ = a'^ in Q and R; find the locus of middle point of P^Q as this line revolves about Py 40. A point moves so that its distance from the point (1, 3) is to its distance from the point (~4, 1) in the ratio 2:3. Find the equation of its locus. 41. Do the circles 4a;2 + 4?/2 + 4x-12?/ + l=0 and 2x^-{-2y^ + y = intersect? Show in two ways. 42. Find the equation of a circle of radius VSS which passes through the points (2, 1) and ("3, 4). 43. What are the equations of the tangent and the normal to the circle x'^ -{- y'^ = 13, — these lines passing through the point (2,-3)? through the point (0, 6) ? 44. Find the equations of the tangents through (2, 3) to the circle 9(a;2+ y^)+Qx-12y + 4. = 0. 100.] THE CIRCLE 169 45. At what angle do the ch-cles ar^ + ^/^ + Oa; — 2y + 5 = and x^ -\- y'^ -\- ^ X -\- 2 y — 5 = intersect each other ? 46. A diameter of the circle 4a;2 + 4?/- + 8x — 12y + l = passes through the point (1, -1). Find its equation, and the equation of the chords which it bisects. 47. Find the locus of a point such that tangents from it to two con- centric circles are inversely proportional to the radii of the circles. 48. Find the locus of a point which moves so that its distances from two fixed points are in constant ratio k. Discuss the locus and draw the figure. 49. A point moves so that the square of its distance from the base of an isosceles triangle is equal to the product of its distances from the other two sides. Show that the locus is a circle. 50. Prove that the two circles x2 + 2/2 + 2 G^x + '2F^y^C^ = and x^ -^ y'^ -\- 2 G^x -\- 2 F,y + C^ = are concentric if (r^ = G^ and F^ = F^] that they are tangent to each other if V(G, - G,y + (F, - F,y =VG,^ + F,' -c,±VG,^ + i'V - ^2 ; and find the condition among the constants that these cu'cles intersect orthogonally, i.e., at right angles to each other. CHAPTER VIII THE CONIC SECTIONS 101. In Art. 48, which should now be carefully re-read, a conic section was defined ; its general equation was de- rived; its three species, viz., the parabola, ellipse, and hyper- bola, were mentioned ; and a brief discussion of the nature and forms of the curve was given. In the present chap- ter, each of these three species will be examined somewhat more closely than was done in Chapter IV, and some general tlieorems concerning its tangents, normals, diameters, chords of contact, and polars will be proved. The general equation (Art. 48) of the conic section might here be assumed, and the special forms for the parab- ola, the ellipse, and the hyperbola be derived from it ; but, partly as an exercise, and partly for the sake of freedom to choose the axes in the most advantageous ways, the equa- tions will here be re-derived, as they are needed, from the definitions of the curves. I. THE PAKABOLA Special Equation of Second Degree Ax^ + 2Goc + 2Ftj+ C = 0, or Bij^ -{-2Chc + 2Fy + C = 102. The parabola defined. A parabola is the locus of a point which moves so that its distance from a fixed point, called the focus, is equal to its distance from a fixed line 170 Ch. VIII. 101-103.] THE CONIC SECTIONS 171 called the directrix. It is the conic section with eccentricity e = 1 (cf. Art. 48). The equation of a parabola, with any given focus and directrix, can be obtained directly from this definition. Example. To find the equation of the parabola whose directrix is the line x-2y-l=0^ and whose focus is the point (2, -3). Let P = (x, y) be any point on the parabola(see Fig. 79) ; then "^-^ is the distance of P from the directrix (Art. 64), + Vs and V(x - 2)^ + (y + 3)2 is the distance of P from the focus (Art. 26); x-2y-l _ hence — — ^-= — - V(a; - 2)2 + {y + 3)2, by definition ; that is, 4x2 + 4a:2/ +^2_i8^_f,962/ + 64 = 0; which is the required equation. The equation obtained in this way is not, however, in the most suitable form from which to study the properties of the curve, but can be simplified by a proper choice of axes. In Art. 48 it was shown that the parabola is symmetrical witli respect to the straight line through the focus and per- pendicular to the directrix, and that it cuts this line in only one point. If this line of symmetry is taken as the 2:-axis, the equation will have no ^-term of first degree [cf. Art. 48, eq. (3)] ; while if the point of intersection of the curve with this axis be taken as origin, the equation will have no con- stant term, since the point (0, 0) must satisfy the equation. With this choice of axes, the equation of the parabola will reduce to a simple form, which is usually called the first standard equation of the parabola. 103. First standard form of the equation of the parabola. Let D'B be the directrix of the parabola, and F its focus ; 172 ANALYTIC GEOMETRY [Ch. VIII. also let the line ZFX, perpendicular to the directrix, be the a:-axis ; denote the fixed distance ZF by 2 p, and let ^ 0, its middle point, be the origin of coordinates; then the line OZ, per- pendicular to OX, is the ^-axis. Let P = (^x, y) be any point on the curve, Fig, 76.^ and draw liQP perpendicular to OY^ also draw the ordinate MP^ and the line FP. The line FP is called the focal radius of P. Then ZO=OF = p, and the equation of the directrix \^ x+p = 0, . . . (1) while the focus is the point (jo, 0). . . . (2) Again, from the definition of the parabola, FP = LP\ [geometric equation] but FP = ^(x-py^y\ and LP=ZO+ OM=p + x ; hence V(a; - pj^ + y'^ = (x +^), whence y^ = ^p^c, . . . [41] which is the desired equation. This first standard form [41] is the simplest equation of the parabola, and the one which will be most used in the subsequent study of the curve. It will be seen later (Chapter XII) that any equation which represents a parab- ola can be reduced to this form. 104. To trace the parabola y^ = 'i.px. From equation [11] it follows : (1) That the parabola passes through the point 0, half way from the directrix to the focus. This point is called the vertex of the curve. (2) That the parabola is symmetrical with regard to the 103-106.] THE CONIC SECTIONS 173 a;-axis ; ^.e., with regard to the line through the focus per- pendicular to the directrix ; this line is called the axis * of the curve. (3) That X has always the same sign as the constant j?, ^.e., that the entire curve and its focus lie on the same side of a line parallel to the directrix, and micfway between the directrix and the focus. ' (4) That X may vary in magnitude from to oo, and when X increases, so also does y (numerically) ; hence the parabola is an open curve, receding indefinitely from its directrix and its axis. The parabola is then an open curve of one branch which lies on the same side of the directrix as does the focus ; when constructed it has the form shown in Fig. 76. 105. Latus rectum. The chord through the focus of a conic, parallel to the directrix, is called its latus rectum. In the figure this chord is R^R. Now WR = 2FR = 'lSR=1ZF=4:p. Hence the length of the latus rectum of the parabola is 4:p; that is, it is equal to the coefficient of x in the first standard equation. 106. Geometric property of the parabola. Second standard equation. Equation [41] may be interpreted as stating an intrinsic property of the parabola, — a property which belongs to every point of the parabola, whatever coordinate axes be chosen. For (see Fig. 76) the equation y^ = 4:px gives the geometric relation or, expressed in words, * The axis of a curve should be carefully distinguished from an axis of coordinates; though they often are coincident lines in the figures to be studied. 174 ANALYTIC GEOMETRY [Ch, VIII. If from any point on the parabola^ a perpendicular is drawn to the axis of the curve^ the square on this perpendicular is equivalent to the rectangle formed by the latus rectum and the line from the vertex to the foot of the perpendicular. This geometric property enables one to write down immedi- ately the equation of the parabola, whenever the axis of the curve is parallel to one of the coordinate axes. E.g.^ if the vertex of the parabola is the point A = (li^ ^), and its axis is parallel to the a^-axis, as in tlie figure, let F be the focus and P = (a;, y) be any point on the parabola ; draw MP perpendicular to the axis AIC. Then MP^ = 4:AF-AM, i.e., (y-k')^ = 4=2^(00-71), . [42] which is the equivalent algebraic equation. This may be taken as a second standard form of the equation, representing the parabola with vertex at the point (A, k'), with axis parallel to the a?-axis, and, if p is positive, lying wholly on the posi- tive side of the line x= h. Equation [42] evidently may be reduced to equation [41] by a transformation of coordinates to parallel axes through the vertex (A, ^), as the new origin. Again, suppose the position of the parabola to be that represented in Fig. 78. The vertex is ^ = (A, A:), and the axis of the parabola is parallel to the ?/-axis. Let P = (x, y') be any point on the curve, and draw MP perpendicular to the axis of the curve. Then MP = 4 AF • AM [geometric property J — 4 (^—p^AM., [since AF is negative] 106-107.] THE CONIC SECTIONS 175 whence, substituting the coordinates of A and P, {x-hf = -^p{y--k), . . . [43] which is another form for the second standard equation of the parabola. Fig. 78. EXERCISES Construct the following parabolas, and find their equations : 1. having the focus at the point (~1, 3), and for directrix the line 3ar-5?/ = 2 (cf. Art. 102); 2. having the focus at the origin, and for directrix the line . 2x-y + ^ = 0', 3. with the vertex at the origin, and the focus at the point (3, 0); 4. with the vertex at the origin, and the focus at the point (0, "3) ; 5. with the vertex at the point (~2, 5), and the focus at the point (-2, 1); 6. with the vertex at the point (~2, -4), and the focus at the point (1, -4) ; 7. having the focus at the point (2p, 0), and for directrix the line x = Q. 8. What is the latus rectum of each of the parabolas of exercises 3 to 6. 9. Describe the effect produced on the form of a parabola by increas- ing or decreasing the length of its latus rectum. 107. Every equation of the form Ax'^ + 2 Goc + 2 Fy + C = O, or By'i -^ 2 Gjc + 2 Fy + C = Oy represents a parabola whose axis is parallel to one of the coordinate axes. Equations [41], [42], and [43] are of the form 176 ANALYTIC GEOMETRY [Ch. VIII. that is, each has one and only one term containing the square of a variable, and no term containing the product of the two variables. Conversely, it may be shown that an equation of either of these forms represents a parabola whose axis is parallel to one of the coordinate axes. A numerical example will first be discussed, by the method which has already been employed in connection with the equation of the circle (Art. 79), and which is applicable also in the case of the other conies. It is the method of reducing the given equation to a standard form, and is analogous to "completing the square" in the solu- tion of quadratic equations. Example. Given the equation 25^2 - 30^ - 50a; + 89 = 0, to show that it represents a parabola ; and to find its vertex, focus, and directrix. Divide both members of the equation by 25, and complete the square of the ^-terms ; the equation may then be written that is, (2/-|)2 = 2(a:-f), whence (2/ - f )^ = 4 • i • (.r - f ) . Now this equation is in the second standard form (cf. equation [42]), and therefore every point on its locus has the geometric property given in Art. 106; and the locus is a parabola. The vertex is at the point (f , I) ; its axis is parallel to the x-axis, extending in the positive direc- tion ; and, since p = 1, its focus is at the point (f^, |), and the directrix is the line x = \^. Consider now the general equation, and apply the same method, taking for example the second form, viz. : Aa^+^ax + 'lFy^- (7=0. Dividing both numbers of the equation by A, completing the square of the a;-terms, and transposing, the equation becomes 107-108.] THE CONIC SECTIONS 177 \ aJ A\f 2AF whence (^+~r) — ^( )\^~ ~~ AJ \ 2AJV 2AF Comparing this equation with the standard equation [43], it is seen that its locus is a parabola, whose axis is parallel to the y-axis, extending in the negative direction if A and IB have like signs, and in the positive direction if A and F have unlike signs. Its vertex is at the point I — -, — XT and, since p = — — — , its focus is at the point ( and its directrix is the line y = a a^-F^- AC A' 2AF 2AF Note. The transformation just given fails if A = ot ii F = 0, for in that case some of the terms in the last equation are infinite. If, how- ever, A=0, the given equation becomes 2 Gx + 2 Fy + e = ; and, this being of the first degree, represents a straight line. If, on the other hand, F=0, the given equation reduces to Ax'^ -{- Gx + C = 0, and repre- sents two straight lines each parallel to the ?/-axis ; they are real and distinct, real and coincident, or imaginary, depending upon the value of G^ — AC. These lines may be regarded as limiting forms of the parab- ola (see Chapter XII). EXERCISES Determine the vertex, focus, latus rectum, equation of the directrix and of the axis for each of the following parabolas ; also sketch each of the figures : 1, i/ - 5x + 4:y -10 =0] 3. 5?/ - 1 = 3?/2 + 4x; 2. 3 a;2 + 12 X + 4 ?/ - 8 = ; 4. y^- -\- 2 y - 12 x - U = 0. 108. Reduction of the equation of a parabola to a standard form. In Art. 102 it was shown that the equation of a parabola having any TAN. AX. GEOM. — 12 178 ANALYTIC GEOMETBT [Ch. VIII. Fig. 79. given directrix and focus is in general not as simple as the standard equa- tion. It will now be shown that if the coordinate axes be transformed so as to be parallel to the axis and directrix of the curve, the equation wiU be reduced to a standard form. For ex- ample, the equation of the parabola with focus at (2, -3), and having for directrix the line x — 2y — 1 = 0, was found to be 4 a;2 + 4 a:^/ + 2/2 - 18 a: + 26 ?/ + 64 = 0. The axis of the curve is a line through (2, ~3) and perpendicular to x-2?/- 1 = 0; its equation is 2 x + y = 1, and it cuts the a:-axis at the angle = tan-i(-2). The point Z is the intersection of the directrix and axis, and may be found from the two linear equations representing these lines ; the vertex A is the point bisecting ZF. If, then, the axes are rotated through the angle ^ = tan-i(-2), the equation will be reduced to the second standard form, [42] ; and if the origin be also removed to the vertex A, the equation will be fiu'ther reduced to the first standard form, [41]. 7 The point Z is (|, -\), A is (If, -f) ; hence, j9 = .4F = -^:, and trans- forming the axes through the angle ^ = tan-i(-^2), to the new origin 9g (|3^-|)^ the equation of the parabola reduces to y'^ =-^a:. V5 The problem of reducing any equation representing a parabola to its standard form is taken up more fully in Chap. XII. EXERCrSES Find, and reduce to the first standard form, the equation of each of the following parabolas ; also make a sketch of each figure : 1. with focus at the point (-1, 3), and having for directrix the line 2. with focus at the point (-8, -|), and having for directrix the line 2a; + 7?/-8 = 0; 3. with focus at the point (a, h), and having for directrix the line a b 108-109.] THE CONIC SECTIONS 179 II. THE ELLIPSE Special Equation of the Second Degree Aqc^ + U2/2 + 2 6?a5 + ^Fy + C = 109. The ellipse defined. An ellipse is the locus of a point which moves so that the ratio of its distance from a fixed point, called the focus, to its distance from a fixed line, called the directrix, is constant and less than unity. The constant ratio is called the eccentricity of the ellipse. This curve is the conic section with eccentricity e<\. (cf. Art. 48.) The equation of an ellipse with any given focus, directrix, and eccentricity may be readily obtained from this definition. Example. An ellipse of eccentricity ^ has its focus at (2, -1), and has the line x + 2y ■= d for directrix. Let P=(x, y) (Fig. 85) be any point on the curve, i^the focus, and PQ the perpendicular from P to the directrix. Then FP = ^QP] but FP = V(a; - 2)2 + {y + l)^, QP = x+^y-^ (Arts. 26, 64), + V5 hence (x - 2y + {y + ly = ^ (x -\- 2y - by-, that is, 41 x^ -Wxy + 29 zf - 140 a; + 170 y + 125 = 0; which is the equation of the given ellipse. As in the case of the parabola, so also here, a particular choice of the coordinate axes gives a simpler form for the equation of the ellipse ; an equation which is more suitable for the study of the curve, and to which every equation representing an ellipse can be reduced. As has been seen in Art. 48, the curve is symmetrical with respect to the line through the focus and perpendicular to the directrix ; and cuts that line in two points, one on either side of the focus. The equation of the ellipse will be in a simpler form if this 180 ANALYTIC GEOMETRY [Ch. VIII. line of symmetry is chosen as the rr-axis, with the origin half way between its two points of intersection with the curve. The resulting equation is the first standard form of the equa- tion of the ellipse. 110. The first standard equation of the ellipse. Let F be the focus, D'D the directrix, and ZFX the perpendicular to BD' through i^, cutting the curve in the two points A' and A (Art. 48)*. Denote by 2 a the length of AA\ and let be its middle point, so that A0= OA' = a. Let ZX be the 2:-axis, the origin, and OY, perpen- dicular to OX, the y-axis. Then, by the definition of the ellipse, AF= eZA, and FA' = eZA' ; . •. AF+FA' = e {ZA -f ZM) = e{ZA + ZA + AA'}, i.e., AA' = e(2ZA+AA'), whence 2 a = 2 \ Z Z' X M "f^ ci-e- \ 1 \ 1 ._._ M J h' y D' Fig B' .81. D', The segment AA' (Fig. 81) of the principal axis inter- cepted by the curve is called its major or transverse axis ; 111-112.] THE CONIC SECTIONS 183 the corresponding segment B' B is its minor or conjugate axis. From the symmetry of the curve with respect to these axes it follows that it is also symmetrical with respect to their intersection 0, the center of the ellipse. It follows also that the ellipse has a second focus at F' = (ae^ 0) C^^^ig- 81) and a second directrix J>'ii>i — the line x — =0 — on the posi- e tive side of the minor axis, and symmetrical to the original focus and directrix, respectively.* The latus rectum of the ellipse, i.e., the focal chord parallel to the directrix (Art. 105), is evidently twice the ordinate of the point whose abscissa is ae. But if Xi = ae^ Vi — ^ "^^1 — ^^ ; or, since h = a VI — e^ y^ — — . Hence the latus rectum is . a a 112. Intrinsic property of the ellipse. Second standard equation. Equation [-14] states a geometric property which belongs to every point of the ellipse, whatever the coordi- nate axes chosen, and to no other point : viz., if P be any point of the ellipse (Fig. 80), then that IS, m words : * To show tills analytically, let OF' = «e, and OZ' — ^f , and let P^{x, y) G be any point on the ellipse, as before. Equation (5), of Art. 110, gives the relation between x and y ; expanding equation (5), and subtracting 4aex from each member, it becomes «2g2 _ 2 aeic + x2 + ?/2 = ^2 _ 2 aex + e^x% which may be written (ae - xy + ?/ = e2 /« _ x\ i.e., ¥P^ = e'^ PL'^ ; which shows that P is on an ellipse whose focus is i^ and whose directrix is D'lDi. \ 2 18i AjSfALYTIC GEOMETRY [Ch. VIII. If from any 'point on the ellipse a perpendicular he draivn to the transverse axis ; then the square of the distance from the center of the ellipse to the foot of this perpendicular^ divided hy the square of the semi-transverse axis, plus the square of the perpendicular divided by the square of the semi-conjugate axis, equals unity. This geometric or physical property belongs to no point not on the curve, and therefore completely determines the ellipse. It enables one to write immediately the equation of any ellipse whose axes are parallel to the coordinate axes. For example : if, as in Fig. 82, the major axis of an ellipse is parallel to the ic-axis, and the center is at the point (7=(A, ^), let P=(x,y) be any point on the curve, and «, 6 be the semi-axes, then that is (ag - h)^ (y -7c)2 1, [45] which is the equation of the given ellipse. Or again, if, as in Fig. 83, the major axis is parallel to the «/-axis ; then, as before CM' Mr ca' cb' 1, 112.] THE CONIC SECTIONS 185 which is the equation of the given ellipse. Equation [45] may be considered a second standard form of the equa- tion of the ellipse ; by a change of coordinates to a set of parallel axes through the center C = (h^ A;), as the new origin, it can be reduced to the first standard form. By Art. 110 the distance from the center of an ellipse to its focus is ae ; but since h^ = a^(l — e^)* [Art. 110, eq. (T)], therefore ae = Va^ — 5^ ; hence, in Figs. 82 and 83, K L Fig. 83. O X Again, the equation of an ellipse, in either standard form, gives the semi-axes as well as the center of the curve, there- fore the positions of the foci are readily determined from either standard form of the equation. EXERCISES Construct the following ellipses, and find their equations: 1. given the focus at the point ( — 1, 1), the equation of the directrix a: — ?/ + 3 = 0, and the eccentricity \ (cf . Art. 109) ; 2. given the focus at the origin, the equation of the directrix x = —Q, and the eccentricity ^ ; * The student should observe that h is the semi-minor-axis and not nec- essarily the denominator of y'^ in the standard forms of the equation of the ellipse — [44], [45], or [46] ; he should also observe that the foci are always on the major axis. 186 ANALYTIC GEOMETRY [Ch. VIII 3. given the focus at the point (0, 1), the equation of the directrix y — 25 = 0, and the eccentricity \ ; 4. given the center at the origin, and the semi-axes V2, VS. Find also the latus rectum. Find the equation of an ellipse referred to its center, whose axes are the coordinate axes, and 5. which passes through the two points (2, 2) and (3, 1). 6. whose foci are the points (3, 0), (~3, 0), and eccentricity \. 7. whose foci are the points (0, 6), (0, -6), and eccentricity |. 8. whose latus rectum is 5, and eccentricity f . 9. whose latus rectum is 8, and the major axis 10. 10. whose major axis is 18, and which passes through the point 6, 4. Draw the following ellipses, locate their foci, and find their equations : 11. given the center at the point (3, ~2), the semi-axes 4 and 3, and the major axis parallel to the a:-axis (cf. Art. 112) ; 12. given the center at the point (~8, 1), the semi-axes 2 and 5, and the major axis parallel to the y-axis ; 13. given the center at the point (0, 7), the origin at a vertex, and (2, 3) a point on the curve ; 14. given the circumscribing rectangle, whose sides are the lines a:-fl = 0, 2a; — 3 = 0, ?/-f6 = 0, 3y-f4 = 0; the axes of the curve being parallel to the coordinate axes. 15. If h becomes more and more nearly equal to a, what curve does the ellipse approach as a limit ? 113. Every equation of the form Anc^ + By^ + ^Goc + ^Fy -f C = O, in which A and B have the same sign, represents an ellipse whose axes are parallel to the coordinate axes. Equations [44], [45], and [46], obtained for the ellipse, are all, when expanded, of the form Ax^ + By'^^-'iax-{-2Fy + Q=0,, . . (1) where A and B have the same sign, and neither of them is zero. Conversely, an equation of this form represents an ellipse 112-113.] THE CONIC SECTIONS 187 whose axes are parallel to the coordinate axes. As in Art. 107, a numerical case will first be examined, and then the general equation taken up in a similar manner. Example. Given the equation 4: x^ + 9 y^ ~ IQ x -\- IS y — 11 = 0, to show that it represents an ellipse, and to find its elements. Completing the square for the terms in x, and also for those in y, and transposing, this equation becomes 4a;2 - 16 a; + 16 + 9 2/2 + 18 y + 9 = 11 + 16 + 9, that is, 4 (a: - 2)2 + 9 (y + 1)2 = 36 ; hence (X - 2)2 (y + iy ^^ 32 22 This equation is of the form [45], and, therefore, its locus has the geometric property given in Art. 112, and is an ellipse. Its center is the point (2, —1); its major axis is parallel to the x-axis, of length 6; its minor axis is of length 4 ; the foci are the points F'=(2-V5,-l), F=(2+V5,-l)r and the equations of the directrices are, respectively, 2 + 9_ x = 2- V5 Following the method illustrated above, of completing the squares, the general equation (1) may be written 188 AJSrALYTIC GEOMETRY [Ch. YIII. that is, Aj V BJ AB which becomes, if the second member be represented by K, "^^ + ^ /^ =l. ... (2) K K A B Comparing this equation with [45] or [46], it is seen to express the geometric relation of Art. 112, and therefore represents an ellipse. Its axes are parallel to the coordinate — J , —^\ and the lengths of the semi-axes are The foci and directrices may be found as above. Note, li A = B, then equation (1) represents a circle (Art. 79). If ABC > BG^ + AF'^, equation (1) having been written with A and B positive, then no real values of x and y can satisfy equation (2), which is only another form of equation (1), and it is said to represent an imaginary ellipse. If ABC = BG^ + AF'^, then x = , and y = are the only real values that satisfy equation (2) ; in that case, this equa- tion is said to represent a point ellipse ; or, from another point of view, (C F\ — — , — - j. Each of the above may be regarded as a limiting form of the ellipse. EXERCISES Determine, for each of the following ellipses, the center, semi-axes, foci, vertices, and latus rectum ; then sketch each curve. 113-114. J THE CONIC SECTIONS 189 1. Sx'^ + d?/ -Qx -27y + 2 = 0. 2. 4:x^ + f -Sx + 2y + 1 =0. 3. x^ +.15 y2 + 4 a; + 60?/ + 15 = 0. 4. By completing the squares of the a:-terms and of the y-terms, and a suitable transformation of coordinates, reduce the equations of exercises 1, 2, and 3 to the standard form [44]. 114. Reduction of the equation of an ellipse to a standard form. It is now evident that, if the directrix and focus of an ellipse are known, as in the example of Art. 109, the transformation of coordinates Fig. 85. which is necessary to reduce the equation to a standard form can easily be determined. To illustrate ; the ellipse of eccentricity f, with focus at F=(2, -I), and having for directrix the line D'D. whose equation is X -{-2i/ = 6, has for its equation (Art. 109) 41 x^ -lQxy + 29 y'^ - 140 a; + 170 y + 125 = 0. Its axis FZ, perpendicular to D'D, has the equation 2 x — y = 5, and cuts the rr-axis at the angle tan-^ 2. If then the coordinate axes are rotated through the angle tan-^2, the equation will be reduced to the second standard form. Again, Z may be found as the intersection of the directrix and axis; it is the point (3, 1). Then A and A', the vertices 190 ANALYTIC GEOMETRY [Ch. VIII. of the ellipse, divide FZ internally and externally in the ratio f ; hence (Art. 30) these coordinates are (-^/, -|), (0, ~5). Also C, the center of the ellipse, is the point (f, ~-^#). If the origin be next transformed to the point C, the equation will be reduced to the first standard form. 12 Since the axis A A' is of length -^, and the eccentricity is |, the semi- 6 V5 axes are — and 2 ; hence the reduced equation, with C as origin and \/5 CA as a;-axis, will be — + ^ = 1. 36 4 The problem of reducing to standard form the equation of an ellipse, when the directrix is not known, will be postponed to Chapter XII. EXERCISES Find, and reduce to the first standard form, the equation of the ellipse : 1. with focus at the point (1, ~o), with the line x + y = 7 for direc- trix, and eccentricity ^; 2. with focus at the point (a, b), the line - + ^ = 1 for directrix, / a 6 and eccentricity — (where Z 1 (cf . Art. 48). 114-116.] THE CONIC SECTIONS 191 Since the hyperbola differs from the ellipse only in the sign of 1 — e\ which is + in the ellipse and — in the hyper- bola, the standard equation of the hyperbola can be derived by the method of Art. 110 ; and it will be found that with choice of axes and notation as there given, the results given in eqs. (1), (2), and (3) of that article apply equally to the hyperbola. If now, since 1 — e^ is negative, the substitution 52 — ^2(^2 _ 1) is made, equation (6) (p. 181) will become or [47] which is the simplest equation of the hyperbola. For variety, this equation will be obtained by a different method. 116. The first standard form of the equation of the hyper- bola. Let F be the focus, D'D the directrix, and e the eccentricity of the curve. Take B'D as the ^/-axis, with the perpendicular OFX upon it, through the focus, as the a;-axis. Let 2p denote the given distance OF^ and let P = (x,y-) be any point of the locus, with coordinates LP and MP. Then but FP = eLP [geometric equation] FP = V(x - 2^)2 + 2/2, and MP = x ; (x— 2^)2 + y2 = A^, that is, (e2 _ 1)2;2 + 2/2 + 4^(92:- 4 jt>2 = 0, . . . (1) which is the equation of the hyperbola referred to its directrix and principal axis as coordinate axes (cf. Art. 48). 192 ANALYTIC GEOMETRY [Ch. VIII. The curve cuts the a:-axis in two points, A = (a;^, 0), and A' = (^x^, 0), — the vertices of the hyperbola, — whose abscissas are determined by the equation (^2 - l)x^ + 4:px + 4:p^ = 0. The abscissa of (7, the middle point of the segment AA\ is, therefore, 0C = ^1±^ = ^^ (Art. 11); 2 e^ —1 hence the center is on the opposite side of the directrix from the focus. Now transform equation (1) to a parallel set of axes through C; the equations for transformation are (Art. 71) x = x' ^ ^^ , and ?/=?/'; substituting these values, and removing accents, eq. (1) becomes which reduces to (^e^ — l')x^ + ^^ = ^ y that is, 4^2g2 4^V ~ * ' ^ ^ (^2 - 1)2 ^2Tri If these denominators are represented by c^ and W' respec- tively, 2.e., if (.2-1) then P = a\e^ - 1), . . . (4) and equation (2) may be written in the simple form a^ = r,,2 i\2 ^ ^^^ ^^ "^ 2 1 ' ' • ' ('^) 116-117.] THE CONIC SECTIONS 193 the standard equation of the hyperbola. Every equation representing an hyperbola can be reduced to this form, as is shown in Chapter XII. The distance from the center to the focus of the hyperbola ^ = 1 is easily found thus : OF =00+ OF g'2_l ' -^ ^2-1' but, from equation (2), a = „^ -. ^ 6;^ — 1 hence OF = ae, therefore the focus F is the point (^ae, 0^. , . . (4) Similarly for the directrix : 00 2p a —f e2 — 1 e hence the directrix is the line x = 0. . . . (5) e As above defined, h is real, and its value is known when a and e are known. In Fig. 86, OB = b, OB' = -b, and h = aVe^-l. 2 2 117. To trace the hyperbola ^-^, = 1. Equation [47] sliows that : (1) The hyperbola is symmetrical with regard to the a;-axis; that is, with respect to the line through the focus and perpendicular to the directrix. This line is therefore called the principal axis of the hyperbola ; (2) The hyperbola is symmetrical with regard to the ?/-axis also; i.e., with regard to the line parallel to the di- rectrix and passing through the mid-point of the segment cut by the curve from its principal axis ; TAN. AN. GEOM. — 13 194 ANALYTIC GEOMETRY [Ch. VIII. (3) For every value of x from — a to a, y is imaginary ; while for every other value of x, y is real and has two values, equal numerically but opposite in sign. But for every value of y, x has two real values, equal numerically and opposite in sign. When x increases numerically from a to 00, then y increases also numerically from to oo. These facts show that no part of the hyperbola lies between the two lines perpendicular to its principal axis and drawn through the vertices of the curve ; but that it has two open infinite branches, lying outside of these two lines. The form of the hyperbola is as represented in Fig. 86. The segment A^A of the principal axis, intercepted by the curve, is called its transverse axis. The segment B^B of the second line of S3anmetry (the «/-axis), where ^'0 = OB — 6, is called the conjugate axis ; and although not cut by the hyperbola, it bears impor- tant relations to the curve. From the symmetry of the hyperbola, with respect to these axes, it follows that it is also symmetrical with re- spect to their intersection 0, the center of the curve. It follows also that there is a sec- ond focus at the point (— ae, 0), and a second directrix in ft the line a: -h - = on the negative side of the conjugate axis, e and symmetrical to the original focus and directrix. (See Art. Ill, foot-note.) The latus rectum of the hyperbola is readily found to be 2 62 a (cf. Arts. 105, 111). 117-118.] THE CONIC SECTIONS 195 118. Intrinsic property of the hyperbola. Second standard equation. Equation [47] states a geometric property which belongs to every point of an hyperbola, whatever the coordi- nate axes chosen, and to no other point ; and which therefore completely defines the hyperbola. With the figure and notation of Art. 117, equation [47] states (Fig. 87) om^mP = 1, a property entirely analogous to that of Art. 112 for the ellipse. It enables one to write at once the equation of an FIG..89. hyperbola with given center and semi-axes, and axes parallel to the coordinate axes. For example, if the transverse axis is parallel to the a;-axis, as in Fig. 88, and the center at the point (7= (A, A;), and a P = (x, y) is any point on the curve ; then CJ? cP 1, a" [48] 196 ANALYTIC GEOMETRY [Ch. VIII. which is the equation of the hyperbola, with a and h as semi- axes. Again, if the transverse axis is parallel to the «/-axis, as in Fig. 89, with the center at the point (7i, A;), the equation of the hyperbola will be found to be Note 1. That the expressions obtained on p. 193 for the distances fi-om the center to the focus and the directrix, of hyperbola [47], are equally true for hyperbolas [48] and [49] follows from the fact that those expressions involve only a, h, and e ; moreover, equation (4) of Art. 116 determines e in terms of a and b ; hence, for all these hyper- bolas, e^ = — — — , the distances from the center to the foci are given by CF=ae=± y/a^TTS and those to the directrices by CZ =~ = ± y/a^ + b^ Note 2. It should be noticed that in equations [47], [48], [49], the negative term involves that one of the coordinates which is parallel to the conjugate axis. EXERCISES 1. Find the equation of the hyperbola having its focus at the point (~1, -1), for its directrix the line Sx — y = 7, and eccentricity f. Plot the curve (cf . Art. 105, and Art. 109, Ex.) . Find the equation of the hyperbola whose center is at the origin and 2. whose semi-axes equal, respectively, 5 and 3 (cf. Art. 116, [47]) ; 3. with transverse axis 8, — the point (20, 5) being on the curve; 4. the distance between the foci 5, and eccentricity V^ ; 5. with the distance between the foci equal to twice the transverse axis. Find the equation of an hyperbola 6. with center at the point (3, -2), semi-axes 4 and 3, and the trans- verse axis parallel to the a;-axis. Plot the curve (cf. Art. 118) ; 118-119.] THE CONIC SECTIONS 197 7. with center at the point (-3, -4), semi-axes 6 and 2, and the transverse axis parallel to the y-axis. Plot the curve. 8. Find the foci and latus rectum for the hyperbolas of exercises 6 and 7. 9. By a suitable transformation of coordinates, reduce the equations of exercises 6 and 7 to the standard form —■ — f-= 1. 10. Find the foci of the hyperbolas ^^ 25 9 ~ ' ^^^ 4 9 '^^^9 4 Plot the curves (/3) and (y). 119. Every equation of the form Ax^ + Bij^ + 2Gx + 2Fy + C = 0, in which A and B have unlike signs, represents an hyperbola whose axes are parallel to the coordinate axes. When cleared of fractions and expanded, the three equations found for the hyperbola are of the form Ax^-{-Bf-{-2ax-{-2F^-\- 0=0, . . . (1) where A aiid B have opposite signs, and neither of them is zero. Conversely, it will now be shown that every equation of this form represents an hyperbola, whose axes are parallel to the coordinate axes. A numerical case will be examined first, and then the general equation. Example. To show that the equation 9 a;^ - 4 ?/2 - 18 a; + 24 ?/ - 63 = represents an hyperbola, and to find its elements. Transposing the con- stant term, and completing the squares of the x-terms and ?/-terms, the equation may be written 9(a;_l)2_4(3/_3)2^36^ Since this equation is of the form [48], its locus has the geometric property given in Art. 118, and therefore represents an hyperbola. Its center is at the point (1, 3), its transverse axis is parallel to the a,--axis, of length 4, and its conjugate axis is of length 6. The eccentricity is e = l Vl3, the foci are at the points (1 - Vl3, 3) and (1 + vT3, 3) ; and the directrices are the lines whose equations are 198 ANALYTIC GEOMETRY [Ch. VIII. Following the method illustrated in the numerical example, the general equation (1) may be written in the form K ' K A B wherein (cf. Art. 113, p. 188), BG^+AF'^-ABO + ^ =^h . . » (2) K= AB Since A and B have opposite signs, the two terms in the first member of this equation are of opposite signs ; the equation is therefore in the form of [48] or [49], and repre- sents an hyperbola. Its axes are parallel to the coordinate axes, its center is the point [ — — , — - ], and its semi-axes are^± — * and ^±-. Note. Since A and B have opposite signs, eqnation (2), which is only another form of equation (1), always represents a real locus ; it is an hyperbola proper except when ABC = BG^ -\- AF\ and it then represents a pair of intersecting straight lines (cf . Art. 67). It is clear that the method shown for the ellipse in Art. 114 can be applied equally well to the hyperbola, to reduce any equation of this curve to the standard form, when the direc- trix is known. The problem of reducing to the standard form the general equation of an ellipse, when the directrix and focus are not known, is considered in full in Chapter XII. * That sign ( + or — ) which makes the fraction positive is to be used. 119-120.] THE CONIC SECTIONS 199 EXERCISES Determine for each of the following hyperbolas the center, semi-axes, foci, vertices, and latus rectum : 1. 16 a:2 - 8 2/2 + 64 a; - 36 2/ + 10 = ; 2. x^-5y^ + 157j -10x-\-l-0; 3. 2x + Qij + Sy^ = x^ + 7. 4. Reduce the equations of exercises 1, 2, 3, to the standard form ^ = 1. Sketch each curve. 120. Summary. In the preceding articles it has been shown that the special equation of the second degree, Ax^ + Bf + 2ax + 2Fy +(7=0, always represents a conic section, whose axes are parallel to the coordinate axes. There are three cases, corresponding to the three species of conic. (1) The parabola : either A or B is zero. In exceptional cases this curve degenerates into a pair of real or imaginary parallel straight lines, and these may coincide. [Art. 107] (2) The ellipse : neither A nor B is zero, and they have like signs. In exceptional cases this curve degenerates into a circle, a point, or an imaginary locus. [Art. 113, Note] (3) The hyperbola : neither A nor B is zero, and they have unlike signs. In exceptional cases this curve degener- ates into a pair of real intersecting lines. [Art. 119] The ellipse and hyperbola have centers, and therefore are called central conies, while the parabola is said to be non- central ; although it is at times more convenient to consider that the latter curve has a center at infinity, on the princi- pal axis (cf. Appendix, Note E). The equation for each conic has two standard forms, which state a characteristic geometric property of the curve, and to which all other equations representing that species can be 200 A^^ALTT1C GEOMETRY [Ch. VIII. reduced. These standard forms are the simplest for stiidy- mg the curves ; l»ut the student must discriminate carefully between general results and those which hold only when the equation is in the standard form. IV. TANGENTS, NORMALS, POLARS, DIAMETERS, ETC. 121. Since the equation Ax'-]- Bf + 2ax-{-2F^-h 0=0 . . . (1) always represents a conic whose axes are parallel to the coordinate axes, and since by giving suitable values to the constants A, B^ (7, F^ and (7, equation (1) may represent any such conic, therefore, if the equations of tangents, normals, polars, etc., to the locus of equation (1) can be found, inde- pendent of the values that A, B^ etc., may have, these equa- tions will represent the tangents, etc., when any special values whatever are given to the constants involved. In the next few articles such equations will be found. 122. Tangent to the conic Ax^ + J52/2 + ^Gx^-'^Fy-\-C = in terms of the coordinates of the point of contact : the secant method. The definition of a tangent has already been given (Art. 81), and the method to be employed here in finding its equation is the one which was used in Art. 84. That article should now be carefully re-read. Let the given conic, z.e., the locus of the equation, Ax' + By''+2ax^-2Fy + C=0, . . . (1) be represented by the curve BMK\ and let P^ = 0^1, y\) be the point of tangency. 120-122.] THE CONIC SECTIONS 201 Through Pi = (a:^, y^) draw a secant line LM, and let Pg^ (^2' ^2) be its other point of intersection with the locus of equation (1). If the point P2 moves along the curve until it comes into coincidence with Pi, the limiting position of the se- cant LMis the tangent PiT. The equation of the line LM is Fig. 90. y - y\=^ -^{X — Xy). (2) X2 Xy If now P2 approaches Pi until x^ = Xi and 7/2 = ?/i, equa- tion (2) assumes the indeterminate form (3) This indeterminateness arises because account has not yet been taken of the path (or direction) by which P2 shall approach Pi, and it disappears immediately if the condition that Pi and P2 are points on the conic (1) is introduced. Since Pi and P2 are on the conic (1), therefore Ax^' + Bij^' + 2 Gx^ ^ 2 F^^ + C = 0, . . . (4) and AX2' + %2' + 2ax2-^2Fi/2 + C=0, . . . (5) Subtracting equation (4) from equation (5), transposing, factoring, and rearranging [cf. Art. 84, equations (8), (9), and (10)], the result may be written y2 — yi ^ M^i + 3^2)+ 2 (^ X2 - xi P(yi + ^0 + 2 P' C^) If this value of ^ ^ is substituted in equation (2), the result is ^ ^ ^ ^'- ^(^i + y2) + 2p(" "^^' " ' ^'^ 202 ^ ANALYTIC GEOMETRY [Ch. VIII. which is the equation of the secant line LM of the given conic (1). If now this secant line be revolved about Pj until P2 comes into coincidence with Pj, i.e., until x.2 = Xi and ^9 = ^1? this equation becomes which is, therefore, the equation of the tangent line PiT at the point Pi. This equation (8) can be put in a much simpler and more easily remembered form, thas : Clearing equation (8) of fractions, and simplifying, it may be written AxiX + B^i7/ + ax-\-F^=Ax^^ + Bi/^^+ax^+Fi/^, ... (9) but, from equation (3), Ax^' + P?/i' + (^x, + Ft/,= - ax, - Fy, - C. hence substituting this value in the second member of equa- tion (9) that equation becomes Ax^x + By^y ^ Gx^-Fy = - Gx,- Fy,- C, . . . (10) and, by transposing and combining, this may be written, ^a5ia5+ 1^2/12/ + G(a?+ici) + 1^(2/ + 1/1) + e = 0.* . . . [50] This is, then, the equation of the tangent to the conic Ax^ -\- By'' ^ "l ax ^ "IFy -\- 0= 0, whatever the values of the coefficients A^ P, 6^, P, and may be ; the point (x,^ y,) being the point of contact. If J. = 0, P=l, a= -2p, F=0 and (7 = 0, then the equa- tion of this conic becomes y^ = 4:px, and the equation of the tangent becomes, yyi = 2p(x + x,); similarly for any other special form of the equation of the conic. * Compare note, Art. 84, ((3). 122-123.] THE CONIC SECTIONS 203 123, Normal to the conic Ax^ + By^ + 2Gx + 2Fy +C=0, at a given point. The normal to a curve has been de- fined (Art. 81) as a straight line perpendicular to a tan- gent, and passing through the point of contact. Therefore, to obtain the equation of a normal to a conic, at a given point on the conic, it is only necessary to write the equation of the tangent to the conic at that point (by Art. 122), and then find the equation of a perpendicular to the tangent which passes through the point of contact (cf. Arts. 53, 62). Example. To find the equation of the normal to the ellipse 18 "^ 8 ~ at the point (3, 2). The equation of the tangent at the point (3, 2) is 3^ 2^_ 18 "^ 8 ~ ' Fig. 91. I.e. 2a; + 3y = 12. The perpendicular line through (3, 2) is which is, therefore, the required normal. Similarly, to find the normal to the conic whose equation Ax''-hB?/^-i-2ax + 2Fi/-hO=0, . . . (1) at the point P^ = (x^, y-^ on the curve. The equation of the tangent at P^ is (Art. 122) Ax^x + By^y + aQc + x^^-V F(^y + y^^+C = . (2) 204 . ANALYTIC GEOMETRY [Ch. VIII. and its slope is, therefore, (Art. 58 (2)) Ax^ -{-G- Hence the required equation of the corresponding normal at Pi is (Arts. 53, 62) EXERCISES 1. Is the line 3 a; + 2 y = 17 tangent to the ellipse 16 x"^ + 25 ^2 = 400 ? 2. Find the equation of a tangent to the conic x'^ + b y'^ — ^ x -{- \0 y -4=0, parallel to the hne y = ^x + 1 (cf. Art. 82). Write the equations of the tangent and normal to each of the follow- ing conies, through a point {x^, y^ on the curve (cf. Art. 122 [50]). 3. -2 + h = 1- 5. x2 = 4p (?/ - 5) ; sketch the figure. 6. 3 x-2 — 5 z/2 + 24 .r = ; sketch the figure. 7. ;r2 4-5 2/2-3a: + 10^-4 = 0; sketch the figure. 8. Derive, by the secant method (cf . Art. 122), the tangent to the parabola y'^ - 4j!)x; the point of contact being {x^, y^. 9. Derive, by the secant method, the tangent to the ellipse a;^ + 4 3/^ - 8x + 20?/ = 0; the point of contact being (x^ y^). Write the equations of the tangents and normals to each of the fol- lowing conies, at the given point; also sketch each figure : 10. 9 a:2 + 5y2 + 36 ;^ + 20 ?/ + 11 = 0, at the point (-2, 1) ; 11. 9 a;2 + 4 ?/2 + 6 X + 4 ?/ = 0, at the point (0, 0) ; 12. / - 6 ?/ - 8a: = 31, at the point ( -3, - 1); * Since the equation of the normal [51] is so readily deduced, in every particular case, from that of the tangent, and since the latter is so easily remembered, it is not recommended that equation [51] be memorized. 123-124.] THE CONIC SECTIONS 205 /-pit flit 13. "4+^=1, at the point (1, V3) ; 14. 3 a:2 + 4 y"^ = 16, at the point (2, "1). 124. Equation of a tangent, and of a normal, that pass through a given point which is not on the conic. The method to be followed in finding the equation of a tangent, or of a normal, that passes through a given point which is not on the conic, may be illustrated by the following example ; the same method is appli- cable to any conic whatever. Let it be required to find the equation of that tangent to the parabola y2_6y _8a,_3i ^0, . . . (1) which passes through the point (~4, -1). This point not being on the parabola, the method of Art. 116 does not apply ; but, assuming for the moment that it is possible to draw such a tangent, let {x^, y-^ be its point of contact. The equation of this tangent is (Art. 122) yi2/-3(y + 2/i)-4(:r + xO-31=0. ... (2) Since this tangent passes through the point (~4, —1), therefore equa- tion (2) is satisfied by the coordinates "~4 and "1, i-e., -?/, -3(-l + ?/i) -4(-4 + x0 -31 =0, . . . (3) which reduces to .r^ + ?/i + 3 = 0. . . . (1) Equation (4) furnishes one relation between the two unknown con- stants x^ and 2/j ; another equation between these two unknowns is fur- nished by the fact that (a-j, y^ is a point on the parabola (1) ; this equation is y,2 _ 62/1 -8a'i- 31 = 0. . . . (5) Solving between equations (4) and (5) gives x^ = - 2± 2 V2 and y^ = -lT 2\/2 ; hence, there are two points on the given parabola the tangents at which pass through the point ("4, -1); their coordinates are (— 2 + 2 a/2, - 1 - 2V2) and (-2-2 \/2, - 1 + 2V2); and substituting either pair of these values for x^ and ^^ in equation (2) gives the equation of a straight line that is tangent to the parabola (1), and that passes through the point (~4, —1). So, too, if it is desired to find the equation of a normal through a point not on the curve, it is only necessary to assume temporarily the coor- dinates of the point on the curve through which this normal passes, and 206 ANALYTIC GEOMETRY [Ch. VIII. then Jind these coordinates by solving two equations, corresponding to equations (4) and (5) above. The problem of finding the above tangent could also have been solved by writing the equation of a line through the point (-4, -1) (Art. 53) and having the undetermined slope ?n, and then so determining jn that the two ]3oints in which this line meets the parabola should be coincident. 125. Through a given external point two tangents to a conic can be drawn. This theorem can be j^roved in precisely the same way as tlie corresponding theorem in the case of the circle (Art. 89) was proved. It may also be proved by the method already applied to the parabola in the preceding article. Let the latter method be adopted. Suppose the equation of the conic to be Ax' + Bi/' -{- 2 ax -}- 2 Ft/ -h O = 0; . . . (1) let the locus of this equation be represented by the curve LPiP2L\ and let Q=(h^ ¥) be the given external point. If Pi = (xi, y{) is a point on IjPiP^U -: then the equa- tion of the tangent at P^ is Ax^x + Byiy ^-G-{x^x-^ + Fiy + y,)+C=0, (2) and this tangent will pass through the point Q if Ahxi + Bky^ -i- G-(Ji-hXi) j^F(k + y,)+C=0, (3) But Pi being on the locus of equation (1), its coordinates Xi and yi also satisfy equation (1) ; i.e., Axi' + %i' + 2axi-{-2Fyi-\-O=0. . . . (4) If now equations (3) and (4) are solved for Xj and j/i, two values of each are found ; these values are both imaginary if Q is within the conic, they are real but coincident if Q is Fig. 93. 124-126.] THE CONIC SECTIONS 20T on the conic, and they are real and distinct if Q i.i outside of the conic. This proves not only the above proposition but also the fact that no real tangent can be draAvn to a conic through an internal point, and that only one tangent can be drawn to a conic through a given point on the curve. 126. Equation of a chord of contact. If the two tangents are drawn from an external point to a conic section, the straight line through the corre- sponding points of tangency is called the chord of contact cor- responding to the point from which the tangents are drawn (cf. Art. 90). Let P^ = (x^^ y^ be the ex- ternal point from which the two tangents are drawn ; T>^= Q^^yd ^^^ ^3 = (^3' ^3)' ti^^ points of tangency of these tangents to the conic whose equation is Ax^-\-By'^-\-2ax-{-2Fy-\-C = 0', .. . . (1) it is required to find the equation of the line through T^ and T^. The equation of the tangent at T^ (cf. Art. 122) is Ax^x-{-By^y + a(x+x^)+F{y-\-y^^^C = (),. . , (2) and the equation of the tangent at T3 is Ax^x-^By^y-\-a(^x-\-x^^ + FQy + y^)+C=0. . . . (3) Since each of these tangents, by hypothesis, passes through Pj, therefore the coordinates x-^ and y^ satisfy both equation (2) and equation (3) ; ^.e., Ax^x^ + By^y^ -f- a(x^ +x,^ + F {y^ + y^^^ C ==0,. . . (4) and Ax^x^ -f By^^y^ + Qi^x^ + H^^F (y^ + y^)^O=0, (5) Y £^ =^ =.k^ ^ m/\ X Fig .93. 208 ANALYTIC GEOMETRY [Ch. VITI. Equations (4) and (5), respectively, assert tliat the points ^2 = (^2' 1/2) ^^^ ^3 = (^3' ^3) are each on the locus of the equation A3Cioc + Bijiy + G(oc + oci)+F(y + yi) + C = 0, . . [52] But equation [52] is of the first degree in the two vari- ables X and ^, hence (Art. 57) its locus is a straight line ; i.e., [52] is the equation of the straight line through T^ and Tg, which was to be found. Note 1. The equation [52] of the chord of contact corresponding to a given external point (a^j, ?/j), and the equation [50] of the tangent whose point of contact is (x^, y^) are identical in form. This might have been expected because the tangent is only a special case of the chord of contact, since the chord of contact, for a given point, approaches more and more nearly to coincidence with a tangent when the point is taken more and more nearly on the curve. Note 2. The present article furnishes another method of treatment for the question of Art. 124. To get the equations of the tw^o tangents that can be drawn through a given external point to a given conic, it is only necessary to write the equation of the chord of contact correspond- ing to this point ; then find the points in w^hich this chord of contact intersects the conic. These are the points of contact of the required tangents, whose equation may then be written down. EXERCISES 1. By first finding the chord of contact (Art. 126) of the tangents drawn from the point (~|, J/) to the conic 4x2 + 2/2 + 24:3; -2y + 17 = 0, find the points of contact, and then write the equations of the tangents to the conic at these points ; verify that these two tangents intersect in the point (-|, V-)- 2. Solve Ex. 1 by the method of Art. 124. 3. Solve Ex. 1 by the method of Art. 89. 4. Find the equation of a normal through the point (7, 5) to the conic 4x2 4- 3/2 + 24x - 2 2/ + 17 = 0. 126-127.] THE CONIC SECTIONS 209 Is it possible to draw more than one normal through (7, 5) to the given conic ? 5. By the methods of Exs. 1, 2, and 3, find the equations of the tangents through the origin to the conic 3 a;2 - 2 ?/^ = 6 a; + 8 y + 6. 6. By the methods of Exs. 1, 2, and 3, find the equations of the tangents through the point ("1, 1) to the conic 9a;2 + 5y2 + 30a; + 20y + 11 = 0. 7. Sketch the conies whose equations are given in Ex. 1, 5, and 6. 8. Find the equations of the tangents to the conic, x^ + 4 y^ = 4, from the point (3, 2). 9. Find the normal to the conic x^ + 4 ?/2 = 4, through the point (3, 2). 10. Solve ExSc 8 and 9, by assuming the slope m of the required line (Art. 53), and then determining m so that the two points in which the line meets the given curve shall be coincident. 127. Poles and polars. If through any given point P^=(x-^^^ ?/j), outside, inside, or on a given conic, a secant is drawn, meeting the conic in two points Q and i?, and if tangents at Q and H are drawn, they will intersect in some point, as P' = (a;', ^'). The locus of P' as the secant revolves about P^ is the polar of the point P-^ (cf. Art. 91) with regard to the given conic ; and P^ is the pole of that locus. To find the equation of the polar of a given point P-^ =Qx^, «/j), with regard to a given conic whose equation is + 6^=0, . . . (1) let QP^R he any position of the secant through P^, and Fig. 91. TAN. AX. GEOM. 14 210 ANALYTIC GEOMETRY [Ch. VIII. let the tangents at Q and R intersect in P' ~ (x\ y'). Then the equation of QP^R (Art. 126) is Ax'x + By'y + a(x + x') + F{y ^y')+O = (2) Since this line passes through P^, therefore the coordinates Xj^ and j/j satisfy equation (2), i.e., Ax^x' + Byy+a(x^-\-x')-{-F(y^-hy')+C = 0,. ..(3) and equation (-3) asserts that the variable point P' = (x\ y'^ lies on the locns of the equation Ax^x+By^y+a(x-\-x;)-\-F(iy^-y{)-{-0 = Q. . . . (4) Equation (4) is of the first degree in the variables x and ?/, hence (Art. 57), its locus is a straight line ; the polar of P^, with regard to the conic (1), i.e.^ the locus of P' ^ is then the straight line whose equation is Ax^oc + Bijiu + G {oc+oc{) + F{y ^y{) + C = 0, . , , [53] Note. That the equation of a tangent [50] and of a chord of con- tact [52] have the same form as equation [53] is due to the fact that a tangent, and a chord of contact, are but special cases of a polar. 128. Fundamental theorem. An important theorem con- cerning poles and polars is : If the polar of the point P^, with . Fig. 95. 127-129.] THE CONIC SECTIONS 211 regard to a given conic, passes through the point P^, then the polar of P^ with regard to the same conic passes through P^. Let the equation of the given conic be Ax^ + By'^ + 2ax + '2Fy + C=Q, . . . (1) and let the two given points be Pi = i^v y\) aiiti P2 - (^2^ ^2)- Then the equation of the polar of P^ with regard to the conic (1) is (Art. 127) Ax^x^By^y^a(x-Vx^)^P{y-Vy^^C==^\ • • • (2) if this line passes through P^, then Ax^x^-^By^y^^aix^^x^^Piy^-^y^-^C^^. . . (3) But the polar of P^ with regard to the conic (1) is Ax^x-vBy^y^a(x^x^-VP{y-^y^)^C=^, • • • (4) and equation (3) shows that the locus of equation (4) passes through the point P^, which proves the proposition. 129. Diameter of a conic section. The locus of the middle points of any system of parallel chords of a given conic is called a diameter of that conic, and the chords which that diameter bisects are called the chords of that diameter. For a given conic, it is required to find the equation of the diameter bisecting a system of chords whose slope is m. Let the equation of the given conic (B.JK, Fig. 96) be Ax^ + By'^^-2ax-\-2Fy + C=0, . . . (1) let the equation of any one of the parallel chords of slope m, LM iov example, be y=7nx-\-h, (2) and let the two points in which it meets the given conic be P\ = (^r Vi) and Pi = (^2' y^)' 212 ANALYTIC GEOMETRY [Ch. VIII. Then (Art. 122, eq. (6)), (3) Fig. 96. If ^ ^ (A, A;) be tlie mid- dle point of the chord P1P2, then = ^l±^andyfc=^^; substituting these values of x-^ H- x^ and ^^ + ^2 ^^^ equa- tion (3), then clearing of fractions and transposing, that equation becomes Ah-^mBk + a-^mF = ^. ... (4) But equation (4) asserts that the coordinates (h, k} of the middle point of any one of this system of parallel chords satisfy the equation Ax + mBi/ + a + niF = 0, . . . [54] which is therefore the equation of the diameter whose chords have the slope m. EXERCISES 1. Find the polar of the point (2, 1) with regard to the hyperbola ^2 _ 2 (?/2 + a;) — 4 = 0. Show that this polar passes through (12, 3), and then verify Art. 128, for this particular case, by showing that the polar of (12, 3), with regard to the given hyperbola, passes through (2, 1). 2. AYrite the equation of the chord of contact of the tangents drawn through (2, 1) to the hyperbola x^ - 2 y'^ - 2 x - i = 0, then find the points in which it meets the curve, get the equations of the tangents at these points, and verify that they pass through the given point (2, 1). 3. By specializing the coefficients in equation [54], prove that the diameter of a circle is perpendicular to the chords of that diameter. 129-130.] THE CONIC SECTIONS 213 Solution. If equation (1) of Art. 123 represents a circle, then A = B, and then equation [54] becomes 1 G + mF y = — X -. , ^ 111 Am i.e., the slope of the diameter is ; but the slope of the given system m of chords is m, hence the diameter is perpendicular to its chords. 4. By means of eq. [54], i.e., by specializing its coefficients, prove that the diameter of a circle passes through the center of the circle. 5. By means of equation [54] prove that any diameter of the ellipse 3 a:2 + ?/'2 — 6 x + 2 ?/ = passes through the center of the ellipse. Does this property belong to all ellipses ? To all conies ? 6. Find the equation of that diameter of the hyiDerbola a;2 - 4 ?/2 + 16 ?/ + 6 a: - 15 = 0, whose chords are parallel to the line ?/ = 2 a; + 10. Does this diameter pass through the center of the curve ? 7. Find the angle between the diameter and its chords in exercise 6. 8. Show that every diameter of the parabola 3 2/^ _ 1(3 x + 12 ?/ = 4 is parallel to its axis. Is this a property belonging to all parabolas ? 9. Derive, by the method of Art. 129, the equation of that diameter of the hyperbola x^ — ^y^ + IQy + Qx — Id = 0, which bisects chords parallel to the line 3 x — 4 ?/ = 12. 130. Equation of a conic that passes through the intersec- tions of two given conies. Let the given conies be S^ = A^x' + B^i/^-2a^x-{-2F^y+C^ = 0, . . . (1) and S,^ = A^x^^B,y^'ia^x + 2F^y+C^_^ = 0', ... (2) then, if k be any constant whatever, S^ + kS^ = ^ . . . (3) represents a conic whose axes are parallel to the coordinate axes (Art. 120), and which passes through the points in which the conies aS'j = and aS'2 = intersect each other (Art. 41); i.e.^ aS'^ + Ar/S'g = represents ?i family of conies, each member of which passes through the intersections of aS'j = and aS'^, = 0. The parameter k may be so chosen that 214 ANALYTIC GEOMETRY [Ch. VIII. the conic (3) shall, in addition to passing through the four points in which 8-^ = and S^ = () intersect, satisfy one other condition ; e.g.^ that it shall pass through a given fifth point. Moreover, if ^^ = and /S'g = are both circles, then aS'j + kS^ = is also a circle (cf. Arts. 95 and 96). V. POLAK EQUATION OF THE CONIC SECTIONS 131. Polar equation of the conic. Based upon the " focus and directrix " definition already given in Art. 48, the polar equation of a conic section is easily derived. Let D' D (Fig. 97) be the given line (the directrix) and the given point (the focus); draw ZOR through and per- pendicular to 2>'i>, and let be chosen as the pole and OR as the initial line. Also let P= (jO, ^) be any point on the locus, and let e be the eccentricity. Draw MP and OK parallel, and LP and fflT perpendicular, to Z>'-Z>, and let OK=l\ then Fig. 97. OP = e • LP, [definition of the curve] = eCZO+ OM); p = el--{-p cos This equation, when solved for ^, may be written in the form P = - 1 , . . . [55] which is the polar equation of a conic section referred to its focus and principal axis ; e being the eccentricity and I the semi-latus-rectum. If e = 1, equation {^55^ represents a parabola ; if e < 1, an ellipse ; and if e > 1, an hyperbola. 130-132.] THE CONIC SECTIONS 215 IN'oTE. Equation [55] shows that if e 1, i.e., if the equation represents an hyperbola, there are two values of 0, viz., = ± cos~'^ (1 :e), for which p becomes infinite. Therefore there are tivo directions in which a line may be drawn to meet an hyperbola at infinity. The three species of conic sections may therefore be distinguished from each other by the number of directions in which lines may be drawn through the focus to meet the curve at infinity. Or, since parallel lines meet at infinity, any point of the plane may be used instead of the focus. 6 COS" 132. From the polar equation of a conic to trace the curve. Suppose e > 1, i.e., suppose equation [55] represents an hyperbola. When ^ = 0, p = , hence p is negative ; as increases, cos decreases, and e cos 1 — e becomes numerically more and more nearly equal to 1 ; therefore p re- mains negative and be- comes larger and larger; p = — CO when 1 — e cos ^ = 0, i.e., when say ; as increases through this value, p becomes + co and then decreases, but remains positive, and becomes equal to I when = 90°; as increases through 90° to 180°, p remains positive, but continues to decrease, reaching its smallest value, viz. p = , when = 180°; as increases from 180° to 270°, p remains . ?- + ^ . I . positive and increases from to I; as increases from 270° to 1 + e 360° — a, p increases from Z to + go ; as ^ increases through 360° — a, p becomes — co ; and finally, as increases from 360° — a to 360°, p re- mains negative, but decreases numerically, reaching the value • again when becomes 360°. ~ Fig. 98. 216 ANALYTIC GEOMETRY [Ch. VIIL These deductions from equation [55] show that the hyperbola has the form represented in Fig. 98, and that, as increases from to a, the lower half .4'TFof the infinite branch at the left is traced ; as increases from a to 360° — a, the right hand branch VA U is traced ; and as 6 in- creases from 360° — a to 360°, the upper half ^.4' of the left hand branch is traced. If increases beyond 360°, the tracing point moves along the same curve ; this is also true if changes from 0° to - 360°. Note. To show the identity of the curve as traced in the present article and in Art. 117, it need only be recalled that e = , and that I = ■ — a a These values substituted above show that a ^ cos-i( ^ -] = tan-i (-), that OA' = - (a + VaH^'), etc. EXERCISES 1. From equation [55] , trace the parabola. 2. From equation [55], trace the ellipse. 3. By means of equation [55], prove that the length of a chord through the focus of a parabola, and making an angle of 30° with the axis of the curve, is four times the length of the latus-rectum. 4. By transforming from rectangular to polar coordinates, derive the polar equations of the conic sections from their rectangular equations. EXAMPLES ON CHAPTER VIM 1. Find the equations of those tangents to the conic 7 x^ -12y^ = 112, which pass through the point ("9, 7). 2. What is the polar of the point (7, 2) with reference to the conic 16?/2 + 9x2 = 144? Find the equation of the line which is tangent to the conic and parallel to this polar. 3. Find the polars of the foci of the ellipse ^+^ = 1, with regard to this ellipse. Also for the parabola y'^ = 4:px. 4. What is the equation of the polar of the center of the conic Ax^ + S?/2 + 2 Gx + 2 F?/ + C = 0, with reference to the conic ? 5. What is the pole of the directrix of the hyperbola a;^ — 4^/^ = 16, with reference to that curve ? 132.] THE CONIC SECTIONS 217 6. The line y = m (x — ae) passes througli the focus of the central conic — i: ^ = 1. On what line does its pole lie? Find the line ioin- ing its pole to the focns. What relation exists between this line and the given focal chord? 7. What is the polar of the vertex of the conic Ax"^ + Bi/ + 2 Gx + 2 Fij + C = Q, ■with reference to the curve ? 8. What is the equation of each common chord of the two conies 16^2+ 9?/2 = lM, 16a:2-9?/2= IM? Hint. Use Art. 130, equation 3 ; find k so that S^ + kS^ can be factored. 9. Prove that the perpendicular dropped from any point of the directrix, to the polar of that point, passes through the focus (a) iovy^ = ^px. (/?) for^±f^=l. Using the simplest standard equations of the conies, find for each 10. the polar of the focus ; 11. the pole of the directrix ; 12. the ratio of the angle subtended by a chord at its pole, and the angle subtended by the same chord at the focus. 13. Find a conic through the intersections of the ellipse 4a:2-|- y'^= 16 and the parabola ?/2=4a: + 4, and also passing through the point 2,. 2. What kind of a conic is it ? 14. Show that the curves h — = 1 and ^ = 1 have the same 16 7 4 5 foci, and that they cut each other at right angles. 15. Find the vertices of an equilateral triangle circumscribed about the ellipse 9 x^ + 16 ^/^ = 144, one side being parallel to the major axis of the curve. 16. Find the normal to the conic 3 x^ + ?/2 — 2 x — ?/ = 1, making the angle tan-^ (f ) with the x-axis. 17. Show that the locus of the pole, with respect to the parabola ?/^ = 4 ax, of a tangent to the hyperbola x'^ — y'^ — a% is the ellipse 4 x^ + ?/2 = 4 a^, 18. Show that — -] ^ = 1, where k is an arbitrarv con- a^ — ^•2 h^ — k'^ stant, represents an ellipse having the same foci as ^ + -^ = 1 when 218 AliALYTIC GEOMETRY [Ch. VIII. 132. k^k'^>b^; given a>6. Determine the nature of the following conies; and also their foci, directrices, centers, semi-axes, and latera recta : 19. ?/2 = (x + 3) (x + 4) ; 20. x^-4:y'^ + x + y + 1=0', 21. x^ = 4:X + ny + 7; 22. 3x2 + / _ 6a; + 8?/ + 1=0; 23. 3 x2 + 5 y = 3 ?/2 + 5 a: ; 24. 9(x^-y) =-dy(l+2x-S y). 25. Show that the polar equation of the parabola, with its vertex at the , . 4»cos^ 26. Show that if the left hand focus be taken as pole, the polar equation of the ellipse is p = — ^^ ^• 1 — e cos d 27. Derive the polar equation of an hyperbola, with its pole at the focus, eccentricity 2, and the distance of the focus from the directrix equal to 6. CHAPTER IX THE PARABOLA 2/2=4 i)x 133. Review. In the preceding cliapter (Arts. 102 to 108), the nature of the parabola has been examined, and its equa- tion derived in two standard forms. These equations are : ^2_ 4,px, if the axis of the curve coincides with the a;-axis, and the tangent at the vertex with the ^-axis ; and (y — ^)2 = 4^ (a; — A), if the axis of the curve is parallel to the a;-axis, and the vertex is at the point (A, ^). In the present chapter, some of the intrinsic properties of the parab- ola are to be studied, i.e., properties which belong to the curve and are entirely independent of the position of the coordinate axes. For this purpose, it will, in general, be easier to use the simplest form of the equation of the curve, viz., 2/^ = 4:px. In every parabola, the value of the eccentricity is e = 1. If the equation of the parabola is ^/^ = 4:px, then the focus is the point (jc», 0), the directrix is the line x = —p, and the axis of the curve is the line ?/ = 0. The equation y^y=2p(x + x^^ represents the polar of the point P^ = (a^j, y^) with respect to the parabola, for all positions of P^. If P^ be outside the curve, this polar is the chord of contact corresj)onding to tangents from P^ ; if P^ be upon the curve, this polar is the tangent at that point. These facts, shown in the 219 220 ANA L YTIC GEOMETR T [Ch. iX. previous chapter, will be assumed in the following dis- cussion. 134= Construction of the parabola. The two conceptions of a locus given in Article 35 lead to tAvo methods for con- structing a curve, viz., by plotting points to be connected by a smooth curve, and by the motion of a point constrained by some mechanical device to satisfy the law which defines the curve. These two methods may be used in constructing a parabola. («) By separate points. Given the focus F and the vertex (7, draw the axis OFX, the directrix D'D cutting this axis in Z^ and also a series of lines perpendicular to the axis at J/^, ilfg, ilfg, etc., respectively. With F as center and ZM^ as radius, describe arcs cut- ting the line at M^ in two points P^ and Q^ ; similarly, with F as center and ZM^^ as radius, cut the line at M^ in P^ and §2 ; ^^^ so on. The points thus found evidently satisfy the definition of the parabola (Art. 102). In this Avay, as many points of the curve as are desired may be found. If these be then connected by a smooth curve, it will be approximately the required parabola (cf. Note B, Appendix). (/3) By a continuously moving point. Let D'B be the directrix and F the focus. Place a right triangle with its longer side KH in coincidence with the axis of the curve, and its shorter side KJ in coincidence with the direc- trix. Let one end of a string of length KH be fastened at 133-135.] THE PARABOLA 221 jff", and the other end at F. If now a pencil point be pressed against the string, keeping it taut while the triangle is moved along the directrix, as indicated in the figure, then, in every position of P, FF = KP, therefore the pencil will trace an arc of a parabola. ^ ^^^ 135. The equation of the tangent to the parabola y^ = ^px in terms of its slope. The equation of a line having the given slope m is y = mx -\-h\ . » o (1) it is desired to find that value of k for which this line will become tangent to the parabola whose equation is y'^ = 4:px. o o . (2) Considering equations (1) and (2) as simultaneous, and eliminating y^ the resulting equation, which is (mx -\'ky^=4:px^ . o „ (3) has for its roots the abscissas of the two points in which the loci of equations (1) and (2) intersect. These roots will become equal (cf. Art, 9), and therefore the points of inter- section will become coincident, if (mk ~^ff — Tf^l^ 0, ^.g., if Therefore A- = ^V m y = tnoc + P in (4) [56] is, for all values of w, the equation of a tangent to the parabola ^2 _ 4 ^^^ 222 ANALYTIC GEOMETRY [Ch. IX. The abscissa of tlie point of contact of the loci of equa- tions (2) and \J)^^ may be found from equation (3), by sub- stituting in it the value of k given in equation (4); it is i^. wr The ordinate may then be found from equation (1); it is —S.. The point of contact is then ( -^, —^\ 136. The equation of the normal to the parabola y^ = ^pqc in terms of its slope. Since, by definition, the normal to a curve is perjDcndicular to the tangent at the point of con- tact, the equation of a normal to the parabola y^ = -ipx . . . (1) is, if m' be the slope of the tangent [Arts. 62, 135], If m be the slope of the normal, then 1 m = „ m and equation (2) may be written y = mx — 2pm — pm^. . . . [57] This is the equation of a normal in terms of its own slope m. 137. Subtangent and subnormal. Construction of tangent and normal. Let Pj = (x^, y^ be any given point on the parabola whose equa- tion is ^2—. i^px. . . . (1) Draw the ordinate MP^, the tangent ^P^ and the normal P^^JV. Fig. 101 135-137.] THE PARABOLA 223 Then by the definitions of Art. 86, the subtangent is TM^ the subnormal is MN^ the tangent length TPy, and the normal length P^N, The tangent at P^ has for its equation (Art. "'^' «/i^ = 2|?(x + a;i), ... (2) hence its a;-intercept is AT = —x-^. But AM=x-^^ therefore TM=2x^. This proves that the subtangent of the parabola y^ = 4:px is bisected at the vertex; and that its length is equal to ttvice the abscissa of the point of contact. The normal at P^ has for its equation (Art. 123) y-yi = -^Q^-^i)-> • • • (3) hence its a;-intercept is AN= x^-{- 2p. But AM= x-^^ therefore M]S/'= 2p. That is, in words, the subnormal of the jjarabola y^ — \px is constant; it is equal to half the latus rectum. These properties of the subtangent and subnormal give two simple methods of constructing the tangent and normal to any parabola at a given point, if the axis of the parabola is given. First method : from the given point, let fall a perpendicu lar P-JKto the axis of the parabola, meeting it in M. The vertex of the curve being at A^ construct the point T on the axis produced, so that TA = AM. The straight line TP^ is the required tangent at P^, and a line through P^ at right angles to this tangent is the required normal. Second method : from the foot of the perpendicular MP^ construct the point iV, so that MN equals twice the distance from vertex to the focus (2 j? = 2 AF^ ; then P^N is the required normal, and a line through P^ at right angles to P^N is the required tangent. 224 ANALYTIC GEOMETRY [Ch. IX. EXERCISES 1. Construct a parabola with focus 2^'^'^ from the dh'ectrix. 2. Construcfc a parabola with latus rectum equal to 6. 3. Find the equations of the two tangents to the parabola y'^ = 4:px, which form with the tangent at the vertex a circumscribed equilateral triangle. Find also the ratio of the area of this triangle to the area of the triangle whose vertices are the points of tangency. 4. Find the equation of a tangent to the parabola y'^ = 4jt?x, perpen- dicular to the line 4 ?/ — a; + 3 = 0, and find its point of contact. 5. Find the equations of the tw^o tangents to the parabola y^ = bx from the point (7, 1), using formula [56]. 6. Write the equations of the tangents to the parabola y"^ = 10 x, at the extremities of the latus rectum. On what line do these tangents intersect? (cf. Art. 138 (5), p. 228.) 7. Write the equations of the tangent and normal to the parabola 2/2 — 9 a;^ at the point (1, 6). 8. Write the equation of the normal to the parabola y^ = Qx, drawn through the point (|, 3). 9. Write the equation of the tangent to the parabola y"^ = ^px, for the point for which the normal length is twice the subtangent; for the point for which the normal length is equal to the difference between the subtangent and subnormal. 10. Two equal parabolas have the same vertex, and their axes are at right angles ; find the equation of their common tangent, and show that the points of contact are each at the extremity of a latus rectum. 11. Find the locus of the middle point of the normal length of the parabola y^ = 4jox. 12. The subtangent of a parabola for the point (5, 4) is 10 ; find the equation of the curve, and length of the subnormal. 13. Find the subtangent, and the normal length, for the point whose abscissa = — 6, and w4iich is on the parabola ?/2 = — 6x. 14. Find the equation of the tangent parallel to the polar of ("1, 2) with respect to the parabola y'^ = \2x\ also find the point of contact, the length of the tangent, and the subtangent. 15. Find the equation of a parabola which is tangent to the line 2y — ?>x — 1, and whose axis is parallel to the x-axis. 137-138.] THE PARABOLA 225 16. Show that the sum of the subtangent and subnormal for any point on the parabola y'^ = 4:px, equals one half the length of focal chord parallel to the corresponding tangent. 17. Show that as the abscissa in the parabola y'^ = ^px increases from to CO, the slope of the tangent diminishes from go to 0; hence the curve is concave toward its axis. < 138. Some properties of the parabola which involve tangents and normals. Let F be the focus, A the vertex, AX the Fig. 103. axis, and i)'2> the directrix of the parabola whose equation is y^ = 4lpx. (1) Through any point P^ = (xyf y^) on the curve draw the tangent TP^^ cutting the ?/-axis in R^ the directrix in S^ and the a:-axis in T\ also draw the normal P^N], the focal chord P^FP^ ; the tangent at P^ ; the lines L^P^ Q and L^P^, per- pendicular to the directrix ; and the lines SF and L^F. Then the following properties of the parabola are readily obtained : TAN. AN. GEOM. 15 226 AyALYTIC GEOMETRY [Ch. IX. (1) The focus is equidistant from the points P^, T^ and N. For ZPj = XjPi = ZA + AM^ = x^+p, TF=TA + AF=x^-\- p, Art. 137 and F]Sr= AM^ + {M^N - AF) = x^-hp; Art. 137 hence FP^=TF=FK The point F is the midpoint of the hypotenuse of the right triangle TP^N, and is therefore equidistant from the vertices T^ P^, and N. Thus a third method is suggested for constructing the tangent and normal at Pj, viz. : by means of a circle, with the focus F as center, and the focal radius FP^ as radius, which cuts the axis in T and N. (2) The tange^it and normal bisect internally and externally^ respectively^ the angle between the focal radius to the point of contact ayid the perpendicular from that point to the directrix. For ZL^P^T=ZP^TF, since L^P^ \\ TF-, and Z TP^F = ZP^ TF, since TF = FP^ ; ZL^P^T = Z.TP^F. Also, Z.FP^N=Z.NP^Q, since P^N'±P^T. (3) Tlirough any point in the plane two taiigents can be dratvn to the parabola (cf. Arts. 89, 125). The line y = mx + ^ (1) m is tangent to the parabola y'^ = Apx for all values of m. If P' = (x\ y'^ be any given point of the plane, then the tan- gent (1) will pass through P' if, and only if, m satisfy the equation , , v y' = mx + =^, m i.e.,ii n^l±2tt:=A^. . . . (2) Therefore two, and only two, values of m satisfy the given conditions; and therefore through any point of the plane two 138.] THE PARABOLA 227 tangents can be drawn to the parabola. If, however, P' is on the curve, then y'^ - ipx' = 0, the two values of m are equal, i.e., the two tangents coincide. If F' is inside the parabola, then ?/'"^ — 4 px' < 0, and the two values of m are imaginary, i.e., there are no real tangent lines. Therefore it is only when F' is outside the parabola that two real and different tangent lines may be drawn from it to the parabola. (4) Thi'ougli any point in the plane three normals can he drawn to the parabola. The line y = mx—2pm — pm^ . . . (1) is normal to the parabola y'^ = 4: px for all values of m (Art. 136). If F' = (x', y') be any point of the plane, then the normal (1) will pass through P' if, and only if, m has a value that will satisfy the equation . y' = x'm — 2 pm — pm^. . . . (2) Since equation (2) is a cubic in m, there are three values of m which satisfy the given conditions, and therefore, in gen- eral, three normals may be drawn to a parabola from a given point. Special cases may, however, arise in which two of the roots of equation (2) are equal, when there would be only two different normal lines ; or all the roots may be equal,* or two imaginary and one real, in both of which cases there would be only one normal line. Through every point at least one normal line can be drawn to the parabola. (5) The tangents at the extremities of a focal chord intersect on the directrix, and at 7'ight angles (cf. (6), below). For, if S= (x\ y') is the point of intersection of the tangents at the extremities of the focal chord, then the chord is the polar of aS^, and its equation is y'y=2p(x + x^). . . . (1) * For only one point, viz. : P' = (2p, 0), are all the roots of equation (2) equal. 228 ANALYTIC GEOMETRY [Ch. IX. But since this line passes through the focus F={p, 0), = 2^:>(^:) H- x') ; i.e., x' = — p. . . . (2) Hence the point P' is on the locus x=—p, i.e., on the directrix. Again, the tangent line passes through the point P' =[ — p, y') if y^ — — mp + — ' i.e., if vi^ + — 771 — 1 = 0. . . . C4) p ^ But the roots of equation (4) are the slopes m' and m" of the two tangents at P^ and P^ ; and by Art. 11, m'm" = — 1. Hence, the tangents at Pi and Pg intersect at right angles. (6) The line joiniyig any point in the directrix to the focus of a pardhola is p)erj)endicular to the chord of contact cor- responding to that point. For ASL,P, = ASFP, since L^Pi = FP^, SP^ is common, Z L,P^S= Z. SP^F; hence, Z SFP, = Z SL,P, = 90°. The property of (5) may now be shown geometrically. Draw the tangent at P2, and suppose it to meet the directrix in aS" ; then, by what has just been proved, Z S' FP2 is a right angle ; then FS' must coincide with FS ; and the tan- gents at Pi and P^ meet on the directrix. 138.] THE PAUABOLA 229 Moreover, Z F2SF1 is 11 right angle, for SPi bisects Z FjSL,, and SF^ bisects Z L,SF. ' (7) J. perpendicular let fall from the focus upon a tangent line meets that tangent upon the tangent at the vertex. For the equation of the tangent at Fi is 1/ig = 2px -^ ^pxi, . . , (1) and the equation of the perpendicular through the focus F = (p, 0) is 2pg == - y^xArpVv • • •' (2) Regarding equations (1) and (2) as simultaneous, and solving to find the point of intersection i?, its abscissa is determined by the equation (4^2 _^ y2^^ j^ p(4:pxi - yf^ = ; or, since g^^ = 4:pxi^ x=0; . , (3) and M is therefore on the tangent at ^. JSToTE. The preceding properties of the parabola have for variety been given in some cases a geometric, in others an analytic, proof. The student is advised to use both methods of proof for each proposition. Other properties of the parabola are given below as exercises for the student, and should be derived by analytic methods. EXERCISES 1. Write the equations of the normals drawn through the point (3, 3) to the parabola y"^ = Q x . 2. The focal distance of any point of the parabola 2/^ = 4:px is p -\- x. 3. The circle on a focal chord as diameter touches the directrix. 4. The angle between two tangents to a parabola is one half the angle between the focal radii of the points of tangency. 230 ANALYTIC GEOMETRY [Ch. IX. 5. The polars of all points on the latus rectum meet the axis of the parabola in the same point ; find its coordinates, for the parabola 6. The product of the segments of any focal chord of the parabola y^ = 4:px equals 7^ times the length of the chord. 7. Two tangents are drawn from an external point P.^^ = (x^, y^ to a parabola, and a third is drawn parallel to their chord of contact. The intersections of the third with each of the other two is half way between Pj and the corresponding point of contact. 8. The area of a triangle formed by three tangents to a parabola is one half the area of the triangle formed by the three points of tangency. 9. The tangent at any point of the parabola will meet the directrix and latus rectum produced, in two points equidistant from the focus. 10. The normal at one extremity of the latus rectum of a parabola is parallel to the tangent at the other extremity. 11. The tangents at the ends of the latus rectum are twice as far from the focus as they are from the vertex. 12. The circle on any focal radius as diameter touches the tangent drawn at the vertex of the parabola. 13. The line joining the focus to the pole of a chord bisects the angle subtended at the focus by the chord. 14. Prove, geometrically, that a perpendicular let fall from the focus upon a tangent line of a parabola meets that tangent upon the tangent drawn at the vertex (cf . (7) of Art. 138, p. 229). 139. Diameters. A diameter lias been defined as the locus of the middle points of a S3^stem of parallel chords. Its equation may be found as follows (cf. Art. 129): Let m be the common slope of a system of parallel chords of the parabola whose equation is ^2 = 4^3^, . . . (1) then the equation of one of these chords is y z= mx H- ^, . . V- (2) 138-139.] THE PARABOLA 231 and the equation of any other chord of the system will differ from this only in the value of the constant term k. The chord (2) meets the parabola (1) in two points and P^ = (x^,y^, and the coordinates of the middle point P' = (x' ^ y') are therefore Fig. 103. ^f^^^i+^, and y^ = y±±^. . 2 (3) Considering (1) and (2) as simultaneous equations, and eliminating x^ it follows that the ordinates of P^ and P^ are the roots of the equation my'^ = 4 j?(y — ^), i.e.^ of y — ^y + -^— = 0. m m (4) Therefore, by Art. 11, ^1 + ^2 = -^' ^•^•' y =~-^ m m hence whatever the value of ^, the coordinates of the middle point of the chord satisfy the equation 2p y=^' m (6) This is, therefore, the equation of the diameter correspond- ing to the system of chords whose slope is w.* * Equation (5) might have been obtained at once as a special form of equation [54], Art. 129, by giving appropriate values to the coefficients A, J5, F, G, and C there used. 232 ANALTTIC GEOMETRY [Ch. IX. 140. Some properties of the parabola involving diameters. The equation of the diameter of the parabola (Art. 139), y-"^, ... (1) m shows at once that every dimneter of the parabola is parallel to the axis of the curve. (See also Ex. 8, p. 213.) Conversely, since any value whatever may be assigned to m, each value deterniining a system of parallel chords, equation (1) may represent any line parallel to the aj-axis, and therefore every line parallel to the axis of a parabola bisects some set of parallel chords^ and is a diameter of the curve. Again, each of the chords cuts the parabola in general in two distinct points, and the nearer these chords are to the extremity of the diameter the nearer are these two points to each other and to their mid-point. In the limiting posi- tion, when the chord passes through the extremity of the diameter, the two intersection points and their mid-point become coincident, and the chord is a tangent. Therefore the tangent at the end of a diameter is parallel to the bisected chords. It follows from the preceding properties, or directly from equation (1), that the axis of the parabola is the only diameter perpendicidar to the tangent at its extremity. The student will readily perceive how the above properties give a method for constructing a diameter to a set of chords, and in particular how to construct the axis of a given parab- ola. Thus the problem of Art. 137, to construct a tangent and normal to a given parabola at a given point, can now be solved even when the axis is not given. If any point on a diameter is taken as a pole, its polar will be one of the system of bisected chords, of slope m. 140-141.] THE PAEABOLA 233 For the pole isP' = [x' ^ — ), hence the equation of its polar (Art 127) is 2 p t.e. y = mx + mx\ which is the equation of a chord of slope m. In other words, the tangents at the extremities of a chord of the parabola inter- sect upon the corresponding diameter. 141. The equation of a parabola referred to any diameter and the tangent at its extremity as axes. In the simplest form of the equation of the parabola, viz., y^ = ^px, . . . (1) the coordinate axes are the principal diameter and the tangent at its extremity. These are the only pair of such lines that are perpendicular to each other (Art. 140). It is now desired to find the equation of the parabola, when referred to any diameter of the curve and the tangent at its extremity as axes. Let any diameter O'X' of the parabola (1) be the new 3:-axis, and the tangent O'Y' at 0' be the new ?/-axis, meeting the old a:-axis at an angle 6. If m = tan 0, . . . (2) then (Art. 135) the coordinates of 0' are -^ and — , and the m^ m equation for transforming the equation from the old axes to a parallel set through the point 0' are (Art. 71), Fio.lOi. ^ = ^''+£-2' 2/ = / + m 2p m Substituting these values in equation (1) gives y2 + ^y^4^^^ (^) (4) 234 ANALYTIC GEOMETRY [Ch. IX. To turn the y-axis to the final position, making an angle $ with the a:-axis, the equations for transformation are (Art. 72, [24]), x' = x" + y" cos 6, y' = y" sin 6, or, by equation (2), x' = x" + —~=, and y' = J^^ o . . (5) V 1 + mP- Vl + w2 Substituting these values in equation (4), it becomes m^ -y 2 = ^p^n . 1 + w^' or, dropping now the accents, which is the required equation of the parabola. This equation may, however, be written more simply. Observing (Art. 103) that JO ( —^ — 1 is the focal distance of the new origm 0', and representing that distance by p', equation (6) becomes ?/2 = 4y:r. . . . [58] This equation is of the same form as equation (1), but is referred to oblique axes. In general, therefore, the equation represents a parabola, and - is the distance of its focus from the origin. Equation [58] states the following property for every point P of the parabola • MP^ = 4:F0''0'M ] a property entirely analogous to that of Arte 106. EXERCISES 1. Find the diameter of y^=—7x, which bisects the chords parallel to the line x — y + 2 = 0. 2. A diameter of the parabola y^ = 8x passes through the point (2, -3) ; what is the equation of its corresponding chords ? 3. Find the equation of the diameter of the parabola ?/2 = 4 a; -f 4 which bisects the chords 2y — 3x = k. 4. Find the equation of the tangent to the parabola (y — 6)2=8(a;+2), which is perpendicular to the diameter ?/ — 4 = 0. 141.] THE PARABOLA ^ 235 5. Show that the pole of any chord is on the diameter which corre- sponds to the chord. 6. What is the equation of the parabola y^ = Sx, when referred to its diameter y — o = and the corresponding tangent as coordinate axes? 7. What is the equation of the parabola (x + 3)^ = 12 (y — 1), when referred to a diameter through the point (3, 4) and the corre- sponding tangent as coordinate axes? 8. Find the pole of the diameter y = k with reference to the parab- ola y^ = 4:px. 9. The polar of any point on a diameter is parallel to the correspond- ing tangent of that diameter. EXAMPLES ON CHAPTER IX Find the equation of a parabola with axis parallel to the a:-axis : 1. passing through the points (0, 0), (3, 2), (3, -2) ; 2. passing through the points (1, 1), ("3, -3), (2, 2) ; 3. through the point (4, "5), with the vertex at the point (3, -7). 4. Find the equation of a parabola through the four points (0, 2), (3,0), (-1,-1), (-3,-2). 5. Find the vertex and axis of the parabola of Ex. 4. Find the equation of a parabola 6. if the axis and directrix are taken as coordinate axes. 7. with the focus at the origin, and the ?/-axis parallel to the directrix. 8. tangent to the line 4:y = Sx — 12, the equation being in the sim- plest standard form. 9. if a focal radius of length 10 lies along the line 4a; — 3?/ — 8 = 0. 10. Two equal parabolas have the same vertex, and their axes are per- pendicular ; find their common chord and common tangent (cf . Ex. 10, p. 224). 11. At what angle do the parabolas of Ex. 10 intersect? 12. Two tangents to a parabola are perpendicular to each other ; find the product of the corresponding sub-tangents. Find the locus of the middle point 13. of all the ordinates of a parabola. 14. of all chords passing through the vertex. 236 ANALYTIC GEOMETRY [Ch. IX. 141. 15. From any point on the latus rectum of a parabola, perpendiculars are drawn to the tangents at its extremities ; show that the line joining the feet of these perpendiculars is a tangent to the parabola. 16. If tangents are drawn to the parabola ^/^ = 4 ax from any point on the line a; + 4fl = 0, their chord of contact will subtend a right angle at the vertex. Two tangents of slope m and m', respectively, are drawn to a parab- ola ; find the locus of their intersection : 17. if mm' = k', 18. if - + — = A' ; 111 m 19. m m' 20. Find the locus of the center of a circle which passes through a given point, and touches a given line. 21. The latus rectum of the parabola is a third proportional to any abscissa and the corresponding ordinate. 22. Find the locus of the point of intersection of tangents drawn at points whose ordinates are in a constant ratio. 23. What is the equation of the chord of the parabola y^ = Sx whose middle point is at (2, -5) ? 24. A double ordinate of the parabola y^ = 4opx is 8/?; prove that the lines from the vertex to its two ends are perpendicular to each other. 25. Find the locus of the center of a circle which is tangent to a given circle and also to a given straight line. 26. Find the intersections of a normal to the parabola wdth the curve, and the length of the intercepted portion. 27. Prove that the locus of the middle point of the normal intercepted between the parabola and its axis is a parabola whose vertex is the focus, and whose latus rectum is one fourth that of the original parabola. 28. Prove that two confocal parabolas, with their axes in opposite directions, intersect at right angles. 29. Find the equation of the parabola when referred to tangents at the extremities of the latus rectum as coordinate axes. 30. The product of the tangent and normal lengths for a certain point of the parabola y^ = 4:px is twice the square of the corresponding ordi- nate ; find the point and the slope of the tangent line. CHAPTER X THE ELLIPSE, ^^ + f- = 1 142. Review. In Chapter VIII the nature of the ellipse has been briefly discussed, and its equation found in the two standard forms : — + ^ = 1, when the axes of the curve are coincident with the coordi- nate axes ; and a? "^ ^2 --L' Avhen the axes of the curve are parallel to the coordinate axes, and the center is the point (A, ¥). In the present chapter it is desired to study some of the intrinsic properties of the ellipse, ^.e., properties which belong to the curve but are independent of the coordinate axes ; and these can for the most part be obtained most easily from the simpler equation, ^4-^ = 1 The ellipse — + 1- = 1 has its eccentricity given bv the ^2 _ 52 relation h"^ = 0^(1 — e^')^ i.e., e^ = — ; its foci are the two points (±ae, 0), and its directrices the lines x = ±^ e (Art. 110). If the axes are equal, so that b = a, the curve takes the special form of the circle, with eccentricity e = 0, 237 238 ANALYTIC GEOMETRY [Ch. X. the two foci coincident at the center, and the directrices infinitely distant. Tlie equation ^ + ^^ = 1 represents the polar of the point (a;^, y-^ with respect to the ellipse ; if the point is outside the curve, this polar line is its chord of contact ; if upon the curve, the polar is the tangent at that point (Arts. 122, 126, 127). These facts will be assumed in the following work. 143. The equation of the tangent to the ellipse ~^+ 2=1 in terms of its slope. The equation of a line having the 2^1 ven slope m is ,7 ^-, ^ ^ ^ y = mx -\- k ; . . . (1) it is desired to find that value of k for which this line will become tangent to the ellipse whose equation is ^ + 1 = 1. ... (2) Considering equations (1) and (2) as simultaneous, and eliminating y, the resulting equation (b^ 4- a^m^^x^ + 2 ahnkx + aV — c^W' = . . (3) determines the abscissas of the two points of intersection of the curves (1) and (2). When the curves are tangent, these abscissas are equal ; therefore and k = ±Vah7i^ -\- b^. Hence y = mx ± ^ cv^rr^ + 6^ . . . [59] is the equation of a tangent to the ellipse — + ^ = 1, for all values of m. 142-144.] THE ELLIPSE 239 Equation [59] shows that there are two tangents to an ellipse parallel to any given line ; and also (Art. 125), that there are two tangents to an ellipse from any external pomt. 144. The sum of the focal distances of any point on an ellipse is constant ; it is equal to the major axis. 2 2 The ellipse ^ + ^ = 1 has its foci at the points a 2 J2 i^i = ( — ae, 0) and F^ = (ae, 0) ; with 52 = «2 _ ^2^2^ ^cf^ ^j.^^ iiQ^ Let Pi=(^Xi, 7/1) be any point on the curve, so that m' = ^'- hW \B' Fig. 105. Then, F^P^ = (x^ 4- ae^ + ^1' = ah^ + 2 aex^ + x^^ + Vi^ = a?e^ + 2 aex^ + ^1' + ^' - ^ a^ I.e., = aV + 2 aex, + ^^'~ ^'^ 2^1^ + a^-aV a" = rt^ + 2 aea^i + ^'^Xi ; 240 AJS^ALYTIC GEOMETRY [Ch. X. Again, F2P1 = (^1 — ct^y + 7/1 = a^e^ — 2 aex^ + x^ + y^^ = a^ — 2 aexi + A^^ i.e.^ F2P1 == a — exy Hence, by addition, F,P, 4- F,F, = 2ai i.e., the sum of the focal distances of any 'point on an ellipse is constant ; it is equal to the major axis. This property gives an easy method of finding the foci of an ellipse when the axes A' A and B'B are given. For F^B + F,B = 2a\ but F,0 = OF2, F2B = F,B=a. Hence, to find the foci, describe arcs with B as center and a = OA as radius, cutting A' A in the points Fi and F2; these points are the required foci. 145. Construction of the ellipse. The property of Art. 144 is sometimes given as the definition of the ellipse ; viz. the ellipse is the locus of a point the sum of whose distances from two fixed points is constant. This definition leads at once to the equation of the curve (cf. Ex. 5, p. 67); and also gives a ready method for its construction. (a) Construction hy separate points. Let A^ A be the given sum of the focal distances, i.e., the major axis of the ellipse ; and Fy and F^ be the given fixed points, the foci. With either focus as center, and with any radius A'R at the center of the ellipse, with the initial side on the major axis, and the terminal side cut- ting the auxiliary circles in R and §, respectively ; and if Y ^^ y. B / / \ /^^ <:;:^ -^s p ^ \ \ a: f / \ \ \ M' J JM j X tz 5^ ^ / y / 1 Fig. lOr. P is the intersection of the abscissa LM with the ordinate MQ, then P is a point on the ellipse. For the coordinates of P are 0M= OQ cos (f> and MP = M'B = OR sin 0, ^.e., X = a cos ^, ^ = ^ sin (f). . . . [60] Now these values satisfy the equation of the ellipse ; for, substituting them in equation (1), gives a^ cos^ 9 • , • 9 • t ^ H 7^ = cos2 (^ -f sm2 <^ = 1 ; hence P is a point of the ellipse. The points P, §, and i? are called corresponding points. The angle <^ is the eccentric angle of the point P;* and the * The eccentric angle of any given point P on an ellipse is readily con- structed thus : produce the ordinate 3IP to meet the major auxiliary circle in Q ; the angle -40^ is the eccentric angle of the point P. 244 ANALYTIC GEOMETRY [Ch. X. two equations [60] are equations of the ellipse in terms of the eccentric angle, for together they express the condition that the point F is on the ellipse (1).* Since, in the figure, A OM' R and OMQ are similar, it follows that MP:MQ= OB: OQ = b:a, and OM': 0M= OB: OQ = h:a; that is, the ordinate of any point on the ellipse is to the ordi- nate of the corresponding p>oint on the major auxiliary circle in the ratio (h : a) of the semi-axes. Similarly for the abscissas of the corresponding points B and P. 147. The subtangent and subnormal. Construction of tan- gent and normal. Let -2 + 7-2 = 1 . . . (1) a'^ 0^ be a given ellipse, then ^ + 1^ = 1, ... (2) a^ 0^ is the tangent to it at a point P-^=(a^j, y^. Let this tangent cut the rr-axis at the point T. Draw the ordinate MPy Then the subtangent is, by definition, TM\ and its numer- ical value is MT=OT- OMi but, from equation (2), 0T= — \ and OM==x^\ a^ hence MT= x tAy-t i.e., TM='^ x^ — c^ * The equations [60] are of great service in studying the ellipse by the methods of the differential calculus. 146-147.] THE ELLIPSE 245 Hence the value of the subtangent, corresponding to any point of the ellipse whose equation is (1), depends only upon the major axis, and the abscissa of the point ; therefore, if a series of ellipses have the same major axis^ tangents drawn to them at the points having a common abscissa will cut the major axis (extended^ in a common point. Fig. 108. This fact suggests a method for constructing a tangent and normal to an ellipse, at a given point : draw the major auxiliary circle ; at Q on this circle, and in MP^ extended, draw a tangent to the circle. This will cut the axis in T \ and Pj Twill be the required tangent of the ellipse at P^. The normal P^iV^may then be drawn perpendicular to P^T. The equation of the normal through P^ is (cf. eq. [51]) OjII y ~ y\ — p^ (^ — ^i) ; therefore the a;-intercept of the normal at that point is ON^ a'^-W' —x-i = e^x a' v 246 ANALYTIC GEOMETRY But the subnormal corresponding to F^ is M]V= ON- OM, and 0M= x^ ; therefore [Ch. X. MN= "^-^x. - X. a'- = x^ = (e^ — l)a7j. Note. From the value of ON it follows that the normal to an ellipse does not, m general, pass through the center, but passes between the center and the foot of the ordinate ; the extremities of the axes of the curve being exceptional points. If, however, a = b, then e = 0, the curve is a circle, and every normal passes through the center (cf. Art. 85). 148. The tangent and normal bisect externally and inter- nally, respectively, the angles between the focal radii of the point of contact. ,EiG.109. f Let the equation of the given ellipse be -^ + t2 = ^ J ^^^^ let -Fj and F^ be the foci, and F^ = (a;^, y^) any given point on the curve. Draw the tangent TF^, the normal P^iV, and also the lines F^F^ and F^F^ W. 147-149.] THE ELLIPSE 247 Then F^N =F^0-\- 0N= ae + A^ [Art. 147] = e(a + ea^j), NF^ = OF^^ - 0N= ae - e\ = e(a — ex^^ , also F^F^ = a + ex^, [Art. 144] and -^2^1 = a — exy Hence F^JST : iVT2 = ^^P^ : F^F^ ; and, by a theorem of plane geometry, this proportion proves that the normal F^JY bisects the angle F^F^F^ between the focal radii. Again, since the tangent is perpendicular to the normal, the tangent F^T will bisect the external angle FoFiW. This proposition leads to a second method of constructing the tangent and normal to an ellipse at a given point (cf. Art. 147). First determine the foci, F^ and F^ (Art. 144), then draw the focal radii to the given point and bisect the angle thus formed, — internally for the normal, externally for the tangent. 149. The intersection of the tangents at the extremity of a focal chord. If P' = (x', y') be the intersection of two tangents to the ellipse a2 "*■ 62 - -^' the equation of their chord of contact is (Art. 126) x'x y'y h^ = 1 (1) If this chord passes through the focus Fo^ (^^5 0), its equation must be satisfied by the coordinates of F^] therefore x'ae _ ^ . ^ ^ -^ = 1, i.e.,x' = -, 248 ANALYTIC GEOMETRY [Ch. X. and the point of intersection P' is on the line, a:= -; i.e., on the directrix corresponding to the focus F^. Similarly, if the chord passes through the focus i^"j = (— ae, U), the poiut P' is on the directrix x = Hence, the tangents at the extremities of a focal chord intersect upon the corresponding directrix. Again, the line joining the intersection P' = ( -, y' ) to the focus has the slope y^- y\_ y' _ ^y' _ ^^y' . m a a (1-6-2) 62 ' ae ^ ^ e while the slope of the focal chord (1) is &V _ 62 ^ a^y' aey' ' hence ra' = -\ m and therefore the line joining the focus to the intersection of the tangents at the ends of a focal chord is perpendicular to that chord. 150. The locus of the foot of the perpendicular from a focus upon a tangent to an ellipse. Let the equation of a tangent to the ellipse (Art. 143), whose equation is ^.+1 = 1. . . . (1) be written in the form y = mx -\- y/ahn^ + h\ . . . (2) Then the equation of a perpendicular to (2), through the focus {ae, 0), is y = (a: — ae), i.e., x + my = ae. . . . (3) If P'=(x', y') is the point of intersection of (2) and (3), it is re- quired to find the locus of P' ; i.e., to find an equation which will be satisfied by the coordinates x', y', whatever the value of m; this must be an equation involving x' and y', but free from m. Since P' is on both lines (2) and (3), therefore y' — mx' = Va^m'^ + b% ... (4) and x' + my' = ae. . . - (5) 149-151.] THE ELLIPSE 2J:9 The elimination of m is accomplished most easily by squaring each member of equations (4) and (5), and adding: this gives (1 + in^) j;'- + (1 + w^) y"^ = ahn^ + a^e^ + h% i.e., (1 + m2)(x'^ +/2) = «2 (^„^2 ^ 1)^ whence, x'^ + jy'^ = a-. Hence, the point P' is on the circle a;2 + ^2 _ ^2. that is, ^Ae /ocus of the foot of a perpendicular from either focus upon a lau' gent to the ellipse is the major auxiliary circle. 151. The locus of the intersection of two perpendicular tangents to the ellipse. Let the equation of any tangent to the ellipse —^-\-^—^ be written in the form (Art. 143) y — mx = 'Va'^m^ + b% . . . (1) then the equation of a perpendicular tangent is m m-^ i.e., my + x = Va^ + 6%w'^. . . . (2) Letting P' = (x', y') be the point of intersection of these two tangents, (1) and (2), it is required to find the locus of P' as m varies in value; that is, to find an equation between x' and y' which does not involve m. Proceeding as in Art. 150 ; since P' is on both lines (1) and (2), therefore y' — mx' = Va^m^ + b% and my' + x' = Va^ + b'^m^. To eliminate m, square both equations, and add : this gives (W2 + 1) y2 + (^^2 ^ 1) ^./2 ^ Qn2 + 1) a2 + ^„^2 + J) ^2^ i.e., x'^ + y'^ = a'^ -\- b^. Therefore, the point of intersection of perpendicular tangents is on the circle a;2 -f y-2 = rt2 + J2^ ^ ^ ^ [-Q1J which is called the director circle for the ellipse. The locim of the inter- section of two perpendicular tanyents to an ellipse is, then, its director circle. 250 ANALYTIC GEOMETRY [Ch. X. EXERCISES 1. Prove that the two tangents drawn to an ellipse from any external point subtend equal angles at the focus. 2. Each of the two tangents drawn to the ellipse from a point on the directrix subtends a right angle at the focus. 3. A focal chord is perpendicular to the line joining its pole to the focus. Show that this is also true for a parabola. 4. The rectangle formed by the perpendiculars from the foci upon any tangent is constant ; it is equal to the square of the semi-minor-axis. 5. The circle on any focal distance as diameter touches the major auxiliary circle. 6. The perpendicular from the focus upon any tangent, and the line joining the center to the point of contact, meet upon the directrix. 7. The perpendicular from either focus, upon the tangent at any point of the major auxiliary circle, equals the distance of the corresponding point of the ellipse from that focus. 8. The latus rectum is a third proportional to the major and minor axes, 9. The area of the ellipse is trah. Suggestion. Employ the fact, proved in Art. 146, that the ordinate of an ellipse is to the corresponding ordinate of the major auxiliary circle as h : a, and thus compare the area of the ellipse with that of its major auxiliary circle. 152. Diameters. As already shown in Articles 129 and 139, the definition of a diameter as the locus of the middle points of a system of parallel chords leads directly to its equation. Let m be the slope of the given system of parallel chords of the ellipse whose equation is $+$=h . (1) and let 2/ = mx + c . , . (2) 151-152.] THE ELLIPSE 251 be the equation of one of these cliords, which meets the curve in the two points P^ = (x^, y{) and P2=(% 2/2)- -^^^ P' = (a;', ^'), be the middle point of this chord, so that , _ ^1 + x^ , _ yx^y% ^ - 2 ' ^ ~ 2 • . (3) / r I / B yp// /// t/^ / //// y/ /A ^ <^ m / \ \ \a X -/ ^\ ^ z / / V^ y 1 ' W / B' Fig .no The coordinates of P^ and P^ ^^® found by solving (1) and (2) as simultaneous equations, therefore the abscissas a^j and x^ are the roots of the equation {c^rr^ + 5^) a;^ + 2 aP'cmx + a^c^ — ^252 _ q^ ^ ^ and the ordinates y-^ and y^ are roots of the equation Hence, by Art. 11, the coordinates of P' are a^rn^ _!_ 52' i' a2^j^2 _|_ 52* • Now, by varying the value of c, equation (6) gives the coordinates of the middle point for each of the chords of the given set. It is required to find the locus of P' for all values of ^ or, y=-—-x\ . . . [62] which is therefore the equation of the diameter bisecting the chords of slope m. The form of equation [62] shows that every diameter of the ellipse passes through the center, 153. Conjugate diameters. Since every diameter passes through the center of the ellipse, and since, by varying tlie slope m of the given set of parallel chords, the corresponding diameter may be made to have any required slope, therefore it follows that every chord which passes through the center of an ellipse is a diameter^ corresponding to some set of parallel chords. In particular, that one of the set of chords given by equation (2), Art. 152, which passes through the center, — ^^g., the chord whose equation is y = mx, ... [63] is a diameter. This diameter bisects the chords parallel to the line [62]; for if m' be the slope of the line [62], then m' = — , a- m hence, nm' = ; • • • [64] a^ 152-154.] THE ELLIPSE 253 and this equation expresses the condition that line [62], which has the slope m\ shall bisect the chords of slope m (Art. 152). But conversely, it expresses also the condition that the line [63] which has the slope m shall bisect the chords of slope wJ . Hence each of the lines [62] and [63] bisects the chords parallel to the other. Hence, if one diameter bisects the chords parallel to a second^ then also the second diameter bisects the chords parallel to the first. Such diameters are called conjugate to each other. Each line of the set of parallel chords in general cuts the ellipse in two distinct points, and the further the chord is from the center, the nearer these two points are to each other, and to their mid-point. In the limiting position, the chord becomes a tangent, with the two intersection points and their mid-point coincident at the point of tangency. Therefore, the tangent at the end of a diameter is parallel to the conjugate diameter. This property, with that of Art. 152, suggests a method for constructing conjugate diameters : first draw a tangent at an extremity of a given diameter (Art. 147), then a line drawn parallel to this tangent through the center of the ellipse is the required conjugate diameter. (See Fig. 111.) 154. Given an extremity of a diameter, to find the extremity of its conjugate diameter. Let P^=(Xp?/j) be an extremity of a given diameter (Fig. ill), then P^=(jx^, -y^ will be the other extremity. Let P/ = (a:/, 3//) and P^ = (jx^, -y{) be the extremities of the conjugate diameter. Let the equation of the ellipse be then the equation of the given diameter P^Pg i^ y=|^, . » « (2) 254 ANALYTIC GEOMETRY [Ch. X. and that of the conjugate diameter Pi'P^j through the center and parallel to the tangent at Pj is 1 7 '^-^ ^^9. * r <^ TNT / / ' / / \ V / ^^jif, ^^^-- — The coordinates of P/ and P{, in terms of Xj, ?/i, a, and &, are given by- equations (1) and (3), considered as simultaneous ; hence, eliminating y between these equations, and remembering that the point P^ is on the ellipse (1) and that therefore If-x^ + d-if' = a^P'^ the abscissas of the points Pj' and Pg' are given by the equation ,2 -t]il> x" = &2 I.e., x{ = -p^ and x^=-^yy Substituting these values in equation (3), gives for the corresponding ordinates, y{=-Xx and y{ = --^r Therefore the required extremities of the conjugate diameter are 155. Properties of conjugate diameters of the ellipse. (a) It has been seen (Art. 153) that two diameters are conjugate when their slopes satisfy the relation mm = -. a"" (1) 154-155.] THE ELLIPSE 255 It follows, since the product of their slopes is negative, that with the exception of the case where one diameter is the minor axis itself, conjugate diameters do not both lie in the same quadrant formed by the axes of the curve. (/S) From the definition (Art. 153) it is evident that the minor and major axes of the ellipse are a pair of conjugate diameters, and they are at right angles to each other. Per- pendicular lines, however, in general, fulfill the condition mm' = — 1 ; . . . (2) hence, in general, equation (2) is not consistent with equa- tion (1) for other values of m and m' than and oo, — the slopes for the axes of the curves. But for b^ — ol^^ ^.e., for the circle, it is clear that every pair of conjugate diameters satisfy equation (2), and are therefore perpendicular to each other. Hence, tlie major and minor axes of the ellipse are the only pair of conjugate diameters that are perpendicular to each other. (7) If, in Fig. Ill, the lengths of the conjugate semi-axes be a^ = CPy, b' = 0P^\ then, since b^x-^ 4- a^y^^ = a%'^^ a''^ = xf + y^., and 6'2 = ^V^; 0^ a'^ therefore a'^ + V^ = ^'^i' +/lf^' + oV+^V = a^ + b^', . . . (3) i.e., the sum of the squares of ttvo conjugate semi-diameters is constant ; it is equal to the sum of the squares of the two semi- axes. 256 ANAL r TIC GEOMETBY [Ch. X. (3) Referring again to Fig. Ill, where C^iV is perpen- dicular to the tangent at Pj, the conjugate diameters .^1^2 and P^^^ intersect at an angle i/r such that -f =Z P^CP{ = 90° +Z P^CN% sin A/r = cos Z P^CN= ^' ^P\ But, by Art. 64, since the equation of the tangent at P^ is IP'x^ + (^y\y — oP-h^^ aH^ ah ah CN = V6V + «Vi^ ^\aH.^ b%^ ^ f ' ^ P ^ a but OP^ = a', ah hence sini/rzz:-— , . . . (4) and the angle hetween tivo conjugate diameters is sin~^——. ah' (e) Tangents at the extremities of a pair of conjugate diameters form a parallelogram circumscribed about the ellipse ; its sides are parallel to, and equal in length to, the conjugate diameters. Since the area of a parallelogram is equal to the product of its adjacent sides and the sine of the included angle, therefore the area of this circumscribed parallelogram is 4:a'h' sin-v^, which, by (4), equals 4 ah. That is, the area of the parallelogram constructed upon any two conjugate diameters is constant; it is equal to the area of the rectangle upon the axes. (f) A simple relation exists between the eccentric angles of the extremities of two conjugate diameters. Let the eccentric angle of Pi= (x-^.y-^ be (/>! (Fig. 112), and of P2 = (^2^ ^2) ^® <^2 j then the slopes of the conjugate diameters may be written (cf. Art. 146), 155-156.] for (7Pi, and for (7^2? But THE ELLIPSE Vi h sin . 4 ' ^^ ^K^i'?', \a ^~^'^ ^7 X 1 ilfj y^ \ ^S^^ \ ^ ~- — -''' Fig US. hence giving W" sin (^1 sin (f)2 _ _^^ a^ cos (^1 cos (^2 ^^ sin_^i_sin_^ _ _ -j ^ cos (^1 COS <^2 that is, sin ^2 sin (f)i + cos (/)2 cos ^^ = 0, whence cos((/)2 — (f>i)= 0. Therefore <^2 - 01 = 90°, and ^Ae eccentric angles of the extremities of two conjugate diameters differ by a right angle. 156. Equi-conjugate diameters. If two conjugate diameters be equal to each other, e.g., if CP^ — CP^ (see Fig. 112), then the properties given in the preceding article lead to other simple onesv Let 0j be the eccentric angle of Pj, then <^j + 90° is the eccentric angle for Pgi hence the coordinates of P^ and Pg ^^^ (^ ^o^ ^v ^ ^^^ 4*i) ^^^ (-a sin (ji^, b cos 0^), and since a'=b', TAN. A>r. GEOM. — 17 258 ANALYTIC GEOMETRY [Ch. X. therefore a^ cos^ ^ + b"^ sin^ <^^ = a^ sin2 <^^ + fts qqq2 ^^^ i.e., taii2 <^i = 1. Hence <^i = 45° or 135° for the extremities of equi-con jugate diameters, and the extremities are p _/ & \ p =/ ^ \ \ a J \ a / The equations of these diameters are b . b y = - X, and y = x. a a Evidently these lines are the diagonals of the rectangle formed on the axes of the curve. By Art. 155, (y), the length of each equi-conjugate semi-diameter is 2 + 62 EXERCISES 1. Find the diameter of the ellipse — + ^- = 1 which bisects the ^ 16 9 chords parallel to the line dx + 5y-\-7 = 0. 2. Find the diameter conjugate to that of exercise 1. 3. Show that the lines 2 x — y = 0, x + dy = are conjugate diame- ters of the ellipse 2 x'^ -\- 3 y^ = 4:. 4. For the ellipse b^x^ + a^y^ = a%\ write the equations of diameters conjugate to the line (a) ax = by, (/?) bx = ay. 5. Prove that the angle between two conjugate diameters is a maximum when they are equal. 6. Show that the pair of diameters drawn parallel to the chords join- ing the extremities of the axes are equal and conjugate. 7. What are the equations of the pair of equi-conjugate diameters of the ellipse lQy^ + 9x^ = 144? 8. Two conjugate diameters of the ellipse h ^ = 1 have the slopes I and — f, respectively; find their lengths. 9. Given the ellipse x'^-^5y^=5, find the eccentric angle for the point whose abscissa is 1. Also find the diameter conjugate to the one passing through this point. 156-157.] THE ELLIPSE 259 10. Given the ellipse 3 a;2 + 4 ?/2 = 12, find the conjugate diameters for the point whose eccentric angle is 30'"\ 11. Find the lengths of the diameters in exercise 10. 12. The lengths of the chord joining the extremities of any two con- jugate diameters of the ellipse ^ + ^ = 1 is Va2+ 62 + a%2sin2<^. Find its greatest value. What is the corresponding value of <^ ? 13. The area of a triangle inscribed in an ellipse, if ^ - cfi^) + sin (<^3 - c^J + sin (l. Subtracting equation (2) from equation (1) gives hence, the difference hetiveen the focal distafices of any point on an hyperbola is constant; it is equal to the transverse axis. If the foci are not given, they may be constructed as follows, provided the semi-axes of the curve are known : plot the points A=(a, 0) and ^ = (0, 6); then with the center, of the hyperbola as center, and the distance AB as radius, describe a circle ; it will cut the transverse axis in the required foci F^ and -Fg, for OF=AB = Va2 + &2 = Va2e2 = ± ae. 162. Construction of the hyperbola. The property of the preceding article might be taken as a new dehnition of the hyperbola, viz. : the hyperbola is the locus of a point the dif- ference of whose distances from two fixed poifits is constant. This definition leads at once to the equation of the curve (cf. Ex. 6, p. 67), and also to a method for its construc- tion. 268 ANALYTIC GEOMETRY [Ch. XI. (a) Construction hy separate points. Let A' A be the given difference of the focal distances, — 2. e., the transverse axis of the hyperbola, — and F^ and F^ the given fixed ^Doints, the foci. With either "-/>, ,'-''V""" focus, say j^i, as a center, j! A R and a radius ^'i^>^'^, F, II describe an arc ; then Fig. 113. --V--' ■-/--' with the other focus as a center, and a radius AH describe an arc cutting the first arcs in the two points Pi. These are points of the hyperbola. Similarly, as many points as desired may be obtained and then connected by a smooth curve, — approximately an hyperbola. (/S) Construction hy a continuously moving point ; the foci being given. Pivot a straight edge LM at one focus P^, so that FtMis greater than the trans- ^ verse axis 2 a ; at iHf and the other ^^-<^^r^ focus F^ fasten the ends of a string ..^l^^^^^^/l of length I, such that F^M=l^2 a ; ^^:^^^^^ AJ I then a pencil P held against the -^ ^ ^ ,,, ^2 ^ ° _ FiG.lU. string and straight edge (see Fig. 114), so as to keep the string always taut, will, while the straight edge revolves about F^, trace one branch of the hyperbola. By fastening the string at the first focus and the straiglit edge at the second, the other branch of the curve can be traced. 163. The tangent and normal bisect internally and exter- nally the angles between the focal radii of the point of contact. Let Fy and F2 be the foci of the hyperbola — — ^ = 1, Fy^ the tangent^ and P^N the normal at the point Pi=(a^i,^i). 162-163.] THE HYPERBOLA 269 Then the equation of Pj^ is ^ - ^ = 1, and the length of ^' the intercept OT oi the tangent is 0T=-, Now, in the triangle F1P1F2, a' FiT=F,0 + 0T= ae +- Xi a and = —(^exi + a), Xi TF,= OF,- 0T = a" ae a?! Xy (^exi — a); [Art. 161] but FiPi = exi 4- ^, and PiF2 = exi — a. Hence F^T : TF^ = F^P, : P1P2. and, by plane geometry, the tangent bisects internally the angle between the focal radii. Then, since the normal is perpendicular to the tangent, the normal PiJV bisects the external angle F2P1W. These facts suggest a method, anal- 270 ANALYTIC GEOMETRY [Ch. XL ogous to that of Art. 148, for constructing the tangent and normal to an hyperbola at a given point. 164. Conjugate hyperbolas. A curve bearing very close relations to the hyperbola = 1 a" is that represented by the equation "P ~ ^ "" ' i.e., by (1) (2) Fig. 116. in which a and h have the same values as in equation (1). This curve is evidently an hyperbola, and has for its trans- verse and conjugate axes, respectively, the conjugate and transverse axes of the original, or primary hyperbola. Two such hyperbolas are called conjugate hyperbolas ; they are sometimes spoken of as the x- and y-hyperbolas, respectively. 163-164.] THE HYPERBOLA 271 It follows at once that the hyperbola (2), conjugate to the hyperbola (1), has for its eccentricity for foci the points (0, ±be')^ and for directrices the lines b Two conjugate hyperbolas have a common center, and their foci are all at the common distance Va^ -j. J2 from this center; i.e., the foci all lie on a circle about the center, having for radius the semi-diagonal OS of the rectangle upon their common axes, and whose sides are tangent to the curves at their vertices. Moreover, when the curves are constructed it will be found that they do not intersect, but are separated by the extended diagonals OS and OK of this circumscribed rectangle, which they approach from opposite sides. These diagonals are examples of a class of lines of great interest in analytic theory, called asymptotes (cf . Art. 3T, (0). EXERCISES 1. Construct an hyperbola, given the distance between its foci as 3 cm. 2o Construct an hyperbola, given the distance from directrix to focus as 2 cm. 3. Write the equation of an hyperbola conjugate to the hyperbola 9x^ — IQy^ = 144, and find its axes, foci, and latus rectum. Sketch the figure. 4. Write the equations of the tangent and normal to the hyperbola 16a;2 _ 9y2 _ j^i2 at the point (4, 4), and find the subtangent and sub- normal. 5. Write the equations of the polars of the point (3, 4) with respect to the hyperbola Qx'^ — IQy^ = 144 and its conjugate, respectively. 272 ANALYTIC GEOMETRY [Ch. XI. 6. For what points of an hj'perbola are the subtangent and sub- normal equal ? 7. Given the hyperbola 9 ?/2 — 4 x^ = 36, find the focal radii of the point whose ordinate is ( "1), and abscissa positive. 8. A tangent which is parallel to the line 5a: — 4?/ + 7 = 0, is drawn to the hyperbola x^ — y'^=Q\ what is the subnormal for the point of con- tact? 9. What tangent to the hyperbola ^ = 1 has its ?/-intercept 2? 10. Find, by equation [67], the two tangents to the hyperbola 4:c2 — 2 2/2 zr 6 which are drawn through the point (3, 5). 11. Find the polars of the vertices of an hyperbola with respect to its conjugate hyperbola. 12. Prove that if the crack of a rifle and the thud of the ball on the target are heard at the same instant, the locus of the hearer is an hj'perbola. 13. An ellipse and hyperbola have the same axes. Show that the polar of any point on either curve is a tangent to the other. 14. Find the equation of an hyperbola whose vertex bisects the dis- tance from the focus to the center. 15. If e and e' are the eccentricities of an hyperbola and its conjugate, then e2 + g'2 ^ g2g'2. 16. If e and e' are the eccentricities of two conjugate hyperbolas, then ae = he'. 17. Find the eccentricity and latus rectum of the hyperbola 2/2 = 4(^2 + a-). 18. Find the tangents to the hyperbola 9 x^ — 16 ?/2 = 144, which, with the tangent at the vertex, form a circumscribed equilateral triangle. Find the area of the triangle. 19. Find the lengths of the tangent, normal, subtangent, and sub- normal for the point (3, 2) of the hyperbola x^ — 2y'^ = 1. 165. Asymptotes. If a tangent to an infinite branch of a curve approaches more and more closely to a fixed straight line as a limiting position, when the point of contact moves further and further away on the curve and becomes infinitely 164-165.] THE HYPERBOLA 273 distant, then the fixed line is called an asymptote of the curve.* More briefly, though less accurately, this defini- tion may be stated as follows : an asymptote to a curve is a tangent whose point of contact is at infinity, but which is not itself entirely at infinity. It is evident that to have an asymp- tote a curve must have an infi- nite branch ; and this branch may be considered as having two coincident, and infinitely distant, points of intersection with its asymptote. This property will aid in obtaining the equation of the asymptote. Y N. ^V N, ^'z s/ \ ■^ / F^ A'V "Va ' F. X / / B^ \ # Fl ^ "% Fig. 117 The hyperbola X a' -2^=1 79 -'-9 (1) (2) is cut by the line y = mx + ^ ft lit ^-|=-1. ... (3) It will be noticed at once that these three equations differ only in their constant terms ; and that the equation of the primary hyperbola (1) differs from that of the asymptotes (2) by the negative of the constant by which the equation of the conjugate hyperbola (3) differs from equation (2). Moreover, this relation between the equations of the three loci must hold when not in their standard forms, z.6., Avhat- ever the coordinate axes. For, any transformation of coor- dinates will affect only the first member of equations (1), (2), and (3), and will affect these in precisely the same way. After the transformation, therefore, the equations of the loci will differ only by a constant (not, however, usually by 1); and the value of the constant in the equation of the asymptotes will be midway between the values of the con- stants in the equations of the two hyperbolas. 276 ANALYTIC GEOMETRY [Ch. XI. Example 1. An hyperbola having the lines (1) 2: + 2^ + 3 = and (2) 3a; + 4^ + 5=0 for asymptotes, will have an equation of the form (a; + 2y + 3)(3a; + 4^ + 5) + A: = 0, . . (3) while the equation of its conjugate hyperbola will be (a; + 2^ + 3)(32; + 4y + 5)-y^ = 0. . . (4) If a second condition is imposed upon the hyperbola, e.g.^ that it shall pass through the point (1, "1), then the value of k may be easily found thus : since tlie curve passes through the point (1, ~1), therefore by equation (3), (l_2 + 3)(3-4 + 5) + A; = 0; .-. ^ = -8, and the equation of the hyperbola is (x + 2y + 3)(3a; + 4?/ + 5) - 8 = 0, that is, 32;2 + 10:?;?/ + 8^2.^142; + 22y + 7 = 0; . (5) and the equation of the conjugate hyperbola is 2>x^ + lOxy + 8^2 ^ 14a: + 22^ + 23 = 0. Example 2. The equation of the asymptotes of the hyperbola 3a;2-14a;^-5i/2 + 7rr + 13^-8 = . . . (1) differs from equation (1) by a constant only, hence it is of the form ^x^-Uxy-by'^^-lx^-ny + k = 0. . . (2) Now equation (2) represents a pair of straight lines, there- fore its first member can be factored, and, by Art. 67, [17] -Wk- ^^ _ 5 i + 24 5 _ 49 ^ = ; i.e., Q4:k = — 384, whence k = — 6, Therefore the equation of the asymptotes is Sx^-Uxy-5y^ + lx + lSy-6 = 0, I.e., (3 a; + 2/ - 2) (^ - 5 ?/ 4- 3) = ; 166-167.] THE HYPERBOLA 277 and the equation of the conjugate hyperbola is 3 :r2 - 14a^^ - 5 ?/2 + 7:^; -f 13 ^ - 4 = 0. 167. Equilateral or rectangular hyperbola. If the axes of an hyperbola are equal, so that a = h, its equation has the foi*!^ x^-y'^ = a\ . . . (1) and its eccentricity e = V2. Its conjugate hyperbola has the equation ^2_^2^_^2. . . , (2) with the same eccentricity and the same shape ; while its asymptotes have the equations x=±y, . . . (3) and are therefore the bisectors of the angles formed by the axes of the curves ; hence the asymptotes of these hyper- bolas are perpendicular to each other. The hyperbola whose axes are equal is therefore called an equilateral, or a rec- tangular hyperbola, according as it is thought of as having equal axes or asymptotes at right angles. EXERCISE 1. Find the asymptotes of the hyperbola 9 a:^ — 16 z/^ = 25, and the angle between them. 2. What are the poles of the asymptotes of the hyperbola 9a;2- 16?/2 = 25 with reference to the curve ? 3. If the vertex lies two thirds of the distance from the center to the focus, find the equations of the hyperbola, and of its asymptotes. 9 4. If a line y = mx + c meets the hyperbola — — ^.^ = 1 in one finite and one infinitely distant point, the line is parallel to an asymptote. 5. Show that, in an equilateral hyperbola, the distance of a point from the center is a mean proportional between its focal distances. 6. Find the equation of the hyperbola passing through the point (0, 7), and having for asymptotes the lines 2x -y = 7, and dx + 3y = b (cf. Art. 166). 278 ANALYTIC GEOMETRY [Ch. XI. 7. Write the equation of the hyperbola conjugate to that of Ex. 6. 8. Find the equations of the asymptotes of the hyperbola 2x^ — xy — 2x = y'^ -}- y -\- Q'^ also find the equation of the conjugate hyperbola. 9. Find the equation of the asymptotes of the hyperbola dx^ + 34:xy + Uy^ - X + 21y = 0. 10. Find the equation of the hyperbola conjugate to 9^2 _ 16^2 _^ 36^^, ^ iQo^ ^ 508. 11. Prove that a perpendicular from the focus to an asymptote of an hyperbola is equal to the semi-conjugate axis. 12. The asymptotes meet the directrices of the x-hyperbola on the a:-auxiliary circle, and of the conjugate hyperbola on the ^/-auxiliary circle. 13. The circle described about a focus, with a radius equal to half the conjugate axis, will pass through the intersections of the asymptotes and a directrix. 14. The line from the center C to the focus F of an hyperbola is the diameter of a circle that cuts an asymptote at P; show that the chords CP and FP are equal, respectively, to the semi-transvei"se and semi- conjugate axes. 168. The hyperbola referred to its asymptotes. If the asymptotes of an hyperbola are chosen as the coordinate axes, their equations will be 2: = and «/ = 0, respectively ; or, combined in one equation, x^ = 0, . . . (1) By the reasoning of Art. 166, it follows that the equation of the. hyperbola, — which differs from that of its asymptotes by a constant, — is x^ = Jc, . . • . (2) wherein the value of the constant 7c is to be determined by an additional assigned condition concerning the curve j e.g., that it shall pass through a given point. 167-168.] THE HYPERBOLA 279 The value of this constant, in terms of a and 6, can in general be found most easily by making the proper trans- formation of coordinates upon the equation of the hyperbola (3) Fig. 118. The new :r-axis makes the angle 6, the new ?/-axis the angle 6\ with the old a;-axis, such that tan 6 = •» tan 0' =-- a a Hence and sm e = -^ine' = -h ■y/a? + 52 cos 6 = + cos 6' = a Va2 + 52 therefore the formulas [24] for transformation, x = x' cos ^ + ?/' COS ^^ ^ = x' sin + y' sin 6\ become in this case x = Va2 + 62 (x' + y ), y = Va2 + 62 (:r'-y), (4) 280 ANALYTIC GEOMETRY [Ch. XI. Appljdng this transformation, equation (3) becomes x'^ + 2 x'y' + y'^ x'^ - 2 x'y' + y'^ ^ ^ , a^ -\- b"^ G^ -\-W' that is, dropping the accents, ^y^ ± - ; a a and the diameters lie on opposite sides of an asymptote. Two conjugate diameters lie in the same quadrant formed hy the axes of the hyperbola^ on opposite sides of the asymptote (cf. Art. 155 (a)). (S) An asymptote passes through the center of an hyper- bola, hence may be regarded as a diameter. Its slope is m= ± -, .*. m = ±-\ a a hence, an asymptote regarded as a diameter is its oivn conju- gate ; it may be called a self-conjugate diameter. This is a limiting case of (7) above. (e) It follows from this last fact that if a diameter inter- sects a given hyperbola, then the conjugate diameter does not intersect it, but cuts the conjugate hyperbola. It is customary and useful to define as the extremities of the conjugate diameter its points of intersection with the conju- gate hyperbola. With this limitation, it follows from (a) of this article, that, as in the ellipse, each of two conjugate diameters bisects the chords parallel to the other. (^) As a limiting case of this last proposition, also, it is evident that the tangent at the end of a diameter is parallel to the conjugate diameter. By reasoning entirely analogous to tliat given in Art. 155, for the ellipse, properties similar to those there given may be derived for the hyperbola. They are included in the following exercises,' to be worked out by the student. 172-173.] THE UYPEEBOLA 287 EXERCISES 1. Find the equation of the diameter of the hyperbola 9 x2 - 16 ?/2 = 25 which bisects the chords y = ?> x -\- h. Find also tlie conjugate diameter. 2. Find, for the liyperbola of Ex, 1, a diameter through the point (1, 1), and its conjugate. X ifi 3. Find the diameter of the hyperbola ^^ ~ nH = 1 which is con- jugate to the diameter x — oy = 0. 4. Find the equation of a chord of the hyperbola 12 x^ — 9 ?/^ = 108, which is bisected at the point (4, 2). 5. Lines from any point of an equilateral hyperbola to the extremi- ties of a diameter make equal angles with the asymptotes. 6. Show that, in an equilateral hyperbola, conjugate diameters make equal angles with the asymptotes. 7. The difference of the squares of two conjugate semi-diameters is constant ; it is equal to the difference of the squares of the semi-axes. 8. The angle between two conjugate diameters is sin-^-^. a'h' 9. The polar of one end of a diameter of an hyperbola, with reference to the conjugate hyperbola, is the tangent at the other end of the given diameter. 10. Tangents at the ends of a pair of conjugate diameters intersect on an asymptote. 173. Supplemental chords. As previously defined, chords of a curve are supplemental when drawn from any point of the curve to the ex- tremities of a diameter. If, in the analytic work of Art. 157, V^ is replaced by —y^, then, if m and m' are the slopes of two supplemental chords of the hyperbola, they must satisfy the relation mm = — . . . . (1) But this is (see Eq. [73]) the condition that exists between the slopes of two conjugate diameters. Therefore, supplemental chords are parallel to a pair of conjugate diameters. For the equilateral hyperbola, i.e., when a = b, this relation has the special value mm' = 1, ... (2) 288 A^^ALYrIC GEOMETRY [Ch. XL and, therefore, the sum of the acute angles which a pair of supplementary chords of the equilateral hyperbola make with its transverse axis is 90° (cf. Art. 172 (/?)). 174. Equations representing an hyperbola, but involving only one variable. (a) Eccentric angle. In the theory of the hyperbola, the auxiliary circles described upon the axes of the curve as diameters are not as useful as the corresponding circles for the ellipse, since the ordinate for a point on the hyperbola does not cut the a:-auxiliary circle, and, there- fore, there is no simple construction for the eccentric angle. It is, how- ever, sometimes desirable to express by means of a single variable the condition that a point shall be on an hyperbola; and for this purpose the equations X = a sec (f), y = b tan =0Q, i.e., OM = a sec <^ ; and, in the right triangle OM'R, M'R = OR tan , i.e., M'R = b tan (jy. But for the point P, x=OM, y = MP = M'R; hence x = a sec <^,.y = h tan <^, and P is a point on the hyperbola.* The eccentric angle for any given point, P, of an hyperbola is easily obtained. Draw the ordinate MP, and from its foot, M, draw a tangent MQ to the a:-auxiliary circle ; then the angle MOQ is the eccentric angle corresponding to P. (/?) The equation of the hyperbola referred to its asymptotes, viz. xy = c^, is satisfied by the coordinates x = ct, y = -, whatever the values of t. The use of this single independent variable t is sometimes convenient in dealing with points on the hyperbola.* EXAMPLES ON CHAPTER XI 1. Write the equation of an hyperbola whose transverse axis is 8, and the conjugate axis one half the distance between the foci. 2. Find tBe equation of that diameter of the hyperbola lQx^ — 9y^ = 14:4: which passes through the point (5, J^) ; also find the coordinates of the extremities of the conjugate diameter. 3. Assume the equation of the hyperbola, and show that the difference of the focal distances is constant. 4. Find the locus of the vertex of a triangle of given base 2 c, if the difference of the two other sides is a constant, and equal to 2 a. 5. Find the* locus of the vertex of a triangle of given base, if the difference of the tangents of the base angles is constant. 6. Find an expression for the angle between any pair of conjugate diameters of an hyperbola. 7. Show that two concentric rectangular hyperbolas, whose axes meet at an angle of 45°, cnt each other orthogonally. * The forms of this article are useful in the differential calculus. TAN. AN. GEOM. 19 290 ANALYTIC GEOMETRY [Ch. XL 8. The portions of any chord of an hyperbola intercepted between the curve and its conjugate are equal. Suggestion. Draw a tangent parallel to the line in question. 9. The coordinates of a point are a tan (0 + a) and b tan (0 + (3) ', prove that the locus of the point, as 6 varies, is an hyperbola. 10. Prove that the asymptotes of the hyperbola a:^ = 3 a: + 5j/ are X = 5 and ^ = 3. 11. If the coordinate axes are inclined at an angle w, find the equa- tion of an hyperbola whose asymptotes are the lines x = 2 and ^ = — 3, respectively, and which passes through the point (2, 1). 12. Find the coordinates of the points of contact of the common tangents to the hyperbolas, a;2 — y2 _ 3 ^2^ ^^^^ ^y — 2a\ 13. If a right-angled triangle be inscribed in a rectangular hyperbola, prove that the tangent at the right angle is perpendicular to the hypothenuse. 14. Show that the line y = mx -{• 2 kV - jn always touches the hyper- bola xy = k^', and that its point of contact is ( , cV- m) . 15. Find the point of the hyperbola xy = 12 for which the subtangent is 4. Find the subnormal for the same point. 16. Find the polar of the point (5, 3) on the hyperbola x^ — 2y'^ = 7^ with respect to the conjugate hyperbola. Show that this line is tangent to the given hyperbola, at the other end of the diameter from (5, 3). 17. If an ellipse and hyperbola have the same foci, they intersect at right angles. 18. Find tangents to the hyperbola 2y^ — IQx^ = 1 which are perpen- dicular to its asymptotes. • 19. Find normals to the hyperbola ^^ ~ ^ v/ ~ -^) = 1 which are -^^ 16 9 parallel to its asymptotes. Find the polar of their point of intersection. 20. Show that, in an equilateral hyperbola, conjugate diameters are equally inclined to the asymptotes. 21. Show that two conjugate diameters of a rectangular hyperbola are equal. 174.] THE HYPERBOLA 291 22. Show that, in an equilateral hyperbola, two diameters at right angles to each other are equal. Show also that this follows from Ex. 21. 23. Find the sum of two focal chords which are, respectively, parallel to two conjugate diameters. 24. Find the common tangents to the hyperbola . = 1 and its mid-circle x^ -\- y^ = ab. ^ 25. In the hyperbola 25 x^ — 16 y^ = 400, find the conjugate diameters that cut each other at an angle of 45°. 26. The latus rectum of an hyperbola is a third proportional to the two axes. 27. The polars of any point (h, k) with respect to conjugate hyperbolas are parallel. 28. The sum of the eccentric angles of the extremities of two conju- gate diameters of an hyperbola is equal to 90°; i.e., cfi -\- cf)' = 90°. 29. Find the equation of a line through the focus of an hyperbola and the focus of its conjugate, and find the pole of that line. 30. Find the asymptotes of the hyperbola xy — 3x — 27/ = 0. What is the equation of the conjugate hyperbola? 31. Show that the ?/-axis is an asymptote of the hyperbola 2xy + 3x^-\-4tx = 9. What is the equation of the other asymptote? Of the conjugate hyperbola ? 32. If two tangents are drawn from an external point to an hyperbola, they will touch the same or opposite branches of the curve according as the given point lies between or outside of the asymptotes. CHAPTER XII GENERAL EQUATION OF THE SECOND DEGREE A3c'^ + 2 Hxij + By- + 2 Gac + 2 Fy -{- C = 175. General equation of the second degree in two variables. Thus far only special equations of the second degree have been studied ; they have all been of the form Ax'^ + Bi^^-\-2Gx + 2Fi/+C=0, . . . (1) ^.e., they have been free from the term containing the product of the variables. In Arts. 107, 113, and 119 it is shown that equation (1) represents a conic section having its axes parallel to the coordinate axes. It still remains to be shown, however, that the most general equation of the second degree, viz. Ax^-\-2ffx7/-{-Bf + 2ax-{-2Fi/-hO=0, ... (2) also represents a conic section. To prove this it is only necessary to show that, by a suitable change of the coordi- nate axes, equation (2) may be reduced to the form of equation (1). If equation (2) be referred to new axes, OX' and OY'^ say, making an angle with the corresponding given axes; and if the new coordinates of any point on the curve be x' and 3/', the old coordinates of the same point being x and ^ ; then (Art. T2) x = x' cos 6 — y' sin ^, and y = x' sin -\- y' cos 6. . . (3) 292 Ch. XII. 175.] EQUATION OF SECOND DEGREE ' 293 Substituting these values (3) in equation (2), it becomes A(x^ cos — 2/' sin Oy + 2 R(x' cos O — y' sin ^) (x' sin 6' + ?/' cos ^) + ^(a;' sin 6 + y' cos (9)2 + 2 (r(a;' cos 6' - ?/' sin 0) -\-2F(x' sin (9 4- ?/' cos 6') + C = 0, . . . (4) which, being expanded and re-arranged, becomes : ^'2(^ cos2 ^ + 2 ^sin ^ cos ^ + 5 sin2 ^) + ^'y'(-2^sin6>cos6'-2^sin2^ + 2^cos2 6'4-25sin^cos6>) + y'XA sin2 0_2 ^sin Oco^e + B cos2 ^) -l-a;'(2(7cos6' + 2^sin(9) + y(-2(7sin(9 + 2^cos6')+ (7=0. ... (5) This transformed equation (5) will be free from the term containing the product x'y' if 6 be so chosen that -2 A sin ^ cos ^ - 2 irsin2 6* + 2 iT'cos^ ^ -f 2 5sin ^ cos6> = 0, i.e., if 2 ^(cos2 - sin2 (9) = (^ - J5)2 sin 6 cos ^, z.e., if 2 ^. cos 2 (9 = (^-^) sin 2 6*, 9 TT or finally, if tan 26 = — ^ — - • . . . (6) Moreover, it is always possible to choose a positive acute angle 6 so as to satisfy this last equation whatever may be the numbers represented by JL, B, and IT. Having chosen 6 so as to satisfy equation (6), and having substituted the values of sin 6 and cos in equation (5), that equation reduces to A'x'^ + B'l/^ + 2 a'x' 4- 2 ^',y ' +(7=0,. . (7) (wherein A\ B\ ••• represent the new coefficients) and therefore represents a conic section with its axes parallel to the new coordinate axes. But equation (7) represents 294 ANALYTIC GEOMETRY [Ch. XII. the same locus as equation (2); hence it is proved that, in rectangular coordinates, every equation of the form represents a conic section whose axes are inclined at an angle 6 to the give7i coordinate axes, zvhere 6 is determined hy the equation ^ 9 tt ^ taii20= "^ A-B It is to be noted that the constant term C has remained unchanged by the transformation given above. The next article will illustrate the application of this method to numerical equations. It is to be observed that this method is entirely general, and enables one to fully determine the conic represented by any given numerical equation of the second degree. Note. In the proof just given that every equation of the second degree represents a conic section, it is assumed that the given axes are at right angles. This restriction may, however, be removed ; for if they are not at right angles, a transformation jnay be made to rectangular axes having the same origin (cf. Arts. T-l, 75), and the equation will have its form and degree left unchanged; after which the proof already given applies. 176. Illustrative examples. Example 1. Given the equation -a:2 + 4:r?/-2/2-4: V2x + 2 V2?/- 11 =0, . . . (1) to determine the nature and position of its locus. Turn the axes through an angle 6, i.e., substitute for x and y, respec- tively, x' cos 9 — y' sin 6 and x' sin -\- y' cos 6] equation (1) then becomes x'2( -cos2 ^ + 4 sin ^ cos - sin2 6) + x'y'{ + 2 sin ^ cos ^ + 4 cos^ ^ - 4 sin2 ^ - 2 sin ^ cos 0) - 2/'2 (sin2 ^ + 4 sin ^ cos ^ + cos2 0) - a;' (4 \/2 cos -2\/2 sin 0) _ + y(+4\/2sin^ + 2 V2cos(9)- 11 =0. ... (2) 175-176.] EQUATION OF SECOND DEGREE 295 The coefficient of x'y' in equation (2) reduces to 4 (sin^ — cos^ 6) ; it will therefore be zero if sin 6 = cos 6, i.e., \i = 45°.* K ^ = 45°, then sin = cos 6 = - — -, and this value of sin and cos V2 being substituted in equation (2), it becomes 3ij'2-2x' + 6y -11 = 0, (3) ■which represents the same locus as is represented by equation (1) ; the difference in the form of the two equations bemg due to the fact that the axes to which equation (3) is referred make an angle of 45° with the axes to which equation (1) is referred. Equation (3) may be written in the form (x'-iy-3(y'-iy = 9, I.e., (x'-iy (y'-iy (V3)^ 32 = 1, (4) which represents an hyperbola (cf. Art. 118). Its center is at the point (1, 1) ; the transverse axis is parallel to the x'-axis; the semi-axes are of length 3 and V3, respec- tively; the eccentricity is ^ e = |V3; the foci are at the points F=(l -t-2 V3, 1) andF' = (l-2V3, 1), re- spectively ; the directrices have the equations x' = 1+ |V3 and 1 - IV3, respectively; and the latus rectum is 2. All these results refer to the new axes, of course, and the locus is that represented in Fig. 121. Fig. 121. * This accords with a result of the preceding article, viz. that to free an equation from its xy-term it is only necessary to turn the axes through a 2 JT positive acute angle determined by tan 2 6 = In the present problem H = -{-2 and A =B = - 1, hence tan 26 = cc and 6 = 45°. 296 ANALYTIC GEOMETRY [Ch. XII. Example 2. Given the equation 4^-2 + 4:c^ + ?/2- 18a; + 26?/ + 64 = 0, ... (5) to determine the nature and position of its locus. Turn the axes through an angle B, i.e., substitute for x and y, respectively, x' cos 6 — y' sin 6 and x' sill 6 -\- y' cosO; equation (.5) then becomes. x'2(4 cos2 + sin2 ^ + 4 sin cos 0) + x'2j'( - 8 cos ^ sin ^ + 2 cos ^ sin 0-4: sin^ ^ + 4 cos'^ 0) + y'-^ (4 shi2 e + cos2^ - 4 sin B cos B) + a;'(-18cos6' + 26sin(9) + ?/'(18sin(9 + 26cos(9)+64 = 0, . . . (6) in which 6 is to be so determined that the coefficient of x'y' shall be zero. On placing this coefficient equal to zero, it is at once seen that tan 2 9=^, from which it follows (cf. exercise 3, Art. 16, second method) that sin 2 ^ = i and cos 2 ^ = | ; remembering that cos 2 ^ = cos^ B — sin^ B = 2 cos^ ^ — 1 = 1 — 2 sin^ B, it is easily deduced that sin 6 = — - and cos B = ^^• Substituting these values in equation (6), it becomes 5x'2 - '2V5x' + 14V5 7/' + 64 = 0, which is the equation of a parabola whose vertex is at the point / 1 63 W5' 14a^5 whose focus is at the point ( — , • |, whose axis coincides with the negative end of the ?/'-axis, and whose latus rectum is — -. All these results refer to the new axes ; the locus of the above equation is given in Fig. 79, p. 178 (Art. 108). EXERCISES 1. For the hyperbola in Fig. 121 find the coordinates of the center and of the foci, and also the equations of its axes and directrices, all referred to the axes OX and OV, 176-177.] EQUATION OF SECOND DEGREE 297 By first removing tlie xy-ievm, determine the nature and position of the loci represented by the following equations. AJso plot the curves. 2. ?/2 _ 2 V3 x^ + 3 a;2 + 6 X - 4 ?/ + 5 = 0. 3. x^ — 4: xy -\- "d y'^ — X — y = Q. 4. 3 :r2 + 2 xi/ + 3 ?/2 - 16 ?/ + 23 = 0. 5. x'^ — 2 xy + y'^ — Q X — Q y -\- d = 0. 177. Test for the species of a conic. It is often desirable to know the species of a conic represented by a given equa- tion even wlien it may not be necessary to determine fully the position of the curve. Remembering that every equa- tion of the second degree represents a conic (Art. 175), and also that the three species of conies may be distinguished from each other by the number of directions in which lines meeting the curve at infinity may be drawn through any given point (Art. 131, Note), it is easy to find a test that will enable one to distinguish at a glance the kind of conic represented by a given equation. Let the given equation be Ax^ + 2Hxy-{-By'^-\-2ax-r'lFy-\-C=0. . (1) If this equation be transformed to polar coordinates, the origin being the pole and the a;-axis the initial line, so that x = p cos 6 and y = p sin ^, it becomes p2(^ cos2 6 + 2ITsmecose + B sin2 6) -{-2p(acose + Fsme}+C=0. . . (2) One value of p, determined by this equation, will be infinite if its direction be such that A cos2 (9 4-2 ^sin 6^ cos 6> + -5 sin^ = 0; [Art» 10] i.e., if B tan2 0-\-2 fftan 6 -\- A = ; I.e., II tan a = — . . . (d) 298 ANALYTIC GEOMETRY [Ch. XII. Equation (3) shows that tan 6 will have two imaginary values, if H^ — AB < ; two real and coincident values, if H^ — AB = ; two real and distinct values, if H^ — AB > 0. Therefore, there is no direction, one direction, or there are two directions, resx^ectively, in which a line meeting the curve in an infinitely distant point may be drawn through the origin, according as R^ - AB is <0, = 0, or > ; and hence, if M^ — AB < 0, equation (1) represents an ellipse, if H^ — AB = 0, equation (1) represents a parabola, if ff^ — AB > 0, equation (1) represents an hyperbola. 178. Center of a conic section. As already defined (Arts. Ill, 117, 120), the center of a curve is a point such that all chords of the curve passing through it are bisected by it. It has also been shown that such a point exists for the ellipse and hyperbola, ^.e., that these are central conies. If the equation of the conic is given in the form Ax^-^2E'x^-{-Bf-{-2ax+2F?/+ 0=0, . (1) the necessary and sufficient condition that the origin is at the center, is (7 = and F = 0. For if the origin be at the center, and (x-^, y^ be any given point on the locus of equation (1), then (^Xy, -y-^ must also be on this locus (because these two points are on a straight line through the origin and equidistant from it) ; hence the coordinates of each of these points satisfy equa- tion (1), i.e., Ax^ + 2 Hx^y^ + By^^ + 2 (^^^^ + 2 F^i + (7 = 0, . (2) 177-179.] EQUATION OF SECOND DEGREE 299 and A(-x^y-\-2R(-x^}(-ij^') + B(^-y,y + ^a^-x,) + 2Fi-y,)+Q=0', (3) • and equation (3) may be written thus : Ax^ + 2 Hx^y^ + By^^ - 2 ai^- 2Fy^-\- C =^. . (4) Subtracting equation (4) from equation (2) gives 4^1 + 4^^1 = 0; i.e., Gx^-\-Fy^ = Q (5) But equation (5) is to be satisfied by the coordinates x-^ and y^ of every point on the locus of equation (1), and the necessary and sufficient conditions for this are a=0 and ^=0. 179. Transformation of the equation of a conic to parallel axes through its center. Let the equation of the given conic be Ax^-\-2Hxy + By'^ + 2ax-\-2Fy+C=0, . (1) and let the coordinates of its center be a and yS. Then to transform equation (1) to parallel axes through the point (a, /3) it is only necessary to substitute in that equation x' -\- a and y^ -\- (3 for x and y. This substitution gives A(x' + a)2 + 2 H(x^ + a)(^' + /5) + B(y^ + py + 2(7(2:' + a)+21^(^''+yS)+ (7=0; i.e., Ax'^ + 2 Hx'y' + By'^ + 2 x'(Aa + ^/3 + 6^) + 2y' {Ha ^BP^-F^ + Aa? + 2 Ha^ + B^ + 2 (7a + 21^/3+ C^* = 0. . . . (2) Since a and ^ are the coordinates of the center (Art. 178), Aa-\-H^^- a = ^ and Ha + BP + F=0', . (3) * It is to be noted here that the new absokite term, le., the term free from x' and y' in equation (2), may be obtained by substituting a and j3 for x and y in the first member of equation (1). 300 ^ ANALYTIC GEOMETRY [Ch. XII. solving these equations gives which are the coordinates* of the center of the locus of equation (1). The constant term in equation (2) is, = a{Aa-\-Hp-\- a) + ^(ffa + B^ + F)-{- Ga + F/3 -h 0, = Cra + Fl3 + 0, [by virtue of equations (3)] . . (5) = Kf^fif )+ <^5|f) + ^' tby equation (4)] ABC+2Faff-AF^-Ba''- OH^ ^ A .g. wherein A=AB0-{-2Faff-AF^-Ba^- OR^ (cf. Art. 67). Equations (4) show that the center of the locus of equa- tion (1) is a definite point, at a finite distance from the origin, if 11^ — AB ^ 0, but that the coordinates of this center become infinite if R^ — AB = 0. Hence (cf . Art. 177), while the ellipse and hyperbola each have a definite finite center, the parabola may be regarded as having a center at infinity. By making use of equations (3) and (5), equation (2) may be written A^'^ + 2ffx'i/' + B^'^-jj~^=0; . . (6) hence, if the general equation of an ellipse or hyperbola be transformed to parallel axes through the center of the conic, the coefficients of the quadratic terms remain unchanged, 179-180.] EQUATION OF SECOND DEGREE ' 301 those of the first degree terms vanish, and the new absolute term becomes ^ ~ if2 _ AB' N'oTE. Two special cases should be noted : 1) Equation (6) shows that if A = 0, the locus of equation (1) con- sists of two straight lines through the new origin (cf. Art. 67). 2) The point (a, ^) is the intersection of the two straight lines Ax ■\- Hij -V G = and Hx + Bij + F = 0. (cf. eq. (3) above.) A JT C If —= — = —, then these' lines are coincident (Art. 38, (ft)), and the coordinates a and /5 become indeterminate. In this case, it may- be shown that A = ; that the locus of equation (1) consists of two lines parallel to, on opposite sides of, and equidistant from, the line Ax -\- Hy + G = ; hence any point of the latter line may be considered as a center, since chords drawn through such a point are bisected by it, i.e., the curve has a line of centers. Again, since W^ — AB = 0, this locus may be considered a special form of a parabola. 180. The invariants A + B and S^ - AB. In Art. 175 it was shown tliat a transformation of coordinates by rotating the axes through an angle 6 changes the coefficients of the equation Ax^ + 2Rxi/ + Bf-\-2ax + 2F^-\-O=0, . (1) with the exception of the constant term. It is true, how- ever, that certain functions of these coefficients are not changed by this transformation, e.g., the sum A -\- B of the coefficients of the x^ and y^ terms is the same after trans- formation as before. If the transformed equation be written A'x^ + 2 ff'xT/ -f- B'f + 2a'x + 2F'y+ Q=0, . (2) wherein, as in Art. 175, A' = A cos2 e + 2 H^in 6 cos 6 -\- B sin^ (9, . (3) 5^ = ^sin2 6>-2^sin6'cos(9 + ^cos2^, . (4) and 2ir' = 2^cos2 6'-(^-^)sin2 6>, . . . (5) 302 ANALYTIC GEOMETRY [Ch. Xll. then the addition of equations (3) and (4) gives A' + B' = A^B (since sin^^ + cos2(9 = 1). . (6) Again, .4' - ^' = 2^sin 2 6" + (^ - ^) cos2 (9 . . (7) hence ^A' - B'y + 4: R'^ = \(A - By -\- -^ R^l (sin2 2 ^ + cos2 2 6), = (A-By + -^II^ ... (8) i.e., A''^ -2A'B' + B'^-]-4:R"^ = A^-2AB + J^ + 4:R^. But by (6), A'^ + 2 A'B' +B'^ = A^ + 2AB-\-B^; hence, by subtraction, H"^-A'B' = S'2-AB, ... (9) and the function R^ — AB is also unchanged by the trans- formation of coordinates, through the angle 6. Moreover, if a transformation of coordinates to a new origin be per- formed as in Art. 179, A, B, and R are not changed, nor, therefore, the functions A + B and R^ — AB. Such functions of the coefficients, which do not vary when the transformations of Arts. 175 and 179 are performed, are called invariants of the equation for those transformations. If, as in Art. 175, 6 be chosen so that tan20 = ^2^, . . . (10) then R' = 0, and equation (8) becomes -A'B' = R'-AB, ... (11) 2 R Again, from eq. (10), sin 2 6 and cos 2 6 = V{A -By + 4. R' A-B -y(A-By+4:R'' hence, equation (8), A' - B' = ^^ . . . . (12) sin 2 6 Since sin 2 ^ is positive (Art. 175), therefore the sign of A' — B' is the same as the sign of R, 180-181.] EQUATION OF SECOND DEGREE 308 These results are useful in reducing an equation of a conic to its simplest standard form, as will be illustrated in the following article. 181. To reduce to its simplest standard form the general equation of a conic, a. Central conic. The result of Art. 180 enables one to reduce to its simplest form a given equation of the second degree, in which H^ — AB^Oy much more easily than by the method of Art. 175. If the equation of the conic, Ax^ + 2 Hxy -i-By^ + 2Gx-\-2Fy + C = 0, (1) be first transformed to the center of the curve as origin, the resulting equation becomes (Art. 179) Ax'^ + 2 Hxy -r By'^ + C = 0. . (2) If equation (2) be now transformed to axes O'Z" and O'F', making the angle 6 with O'X' and O'Y'y respectively, such that A-B it will become (Art. 175) J['x2-f5y+C"=0, (3) wherein the new coeffi- cients are easily deter- mined by the relations C'= Ga + F/3+ C A h-2-ab' (Art. 179), A' + B'=A +B, and -A'B' = H^-AB (Art. 180). Example. Suppose the given equation to be 3x2 + 2x?/ + 3?/2- 16y + 20 = 0, ^ , in which A = 3, H = 1, B == ^, G = 0, F= -8, and C = 20. Then H^ — AB = — 8, and the locus is an ellipse. Fig. 122. (4) 804 ANALYTIC GEOMETRY [Ch. XII. The coordinates of the center are a=: — 1, ^8 = 3. Therefore, C' = &'a + F^+ C= -4; .4' + 5' = 6, -A'B'=-^; and, since A' is larger than B', H being positive (Art. 180), hence ^' =4, 5' = 2; while tan 2 ^ = co , and therefore ^ = 45°. The transformed equation is therefore 4x2 + 2 2/2-4 = 0, O 9 I.e. Y + | = i, ... (5) when referred to the axes 0'X'\ 0' Y" ; and the locus is approximately as given in Fig. 122. h. Non-central conic. If H'^ — AB = 0, the relations of equations (6) and (11), Art. 180, may still be used to simplify the reduction of equa- tion (1) to the standard form for the equation of a parabola, if, as in Art. 176, the xy-tex\x\. be removed first. In this case, however, a better method of reduction is as follows : Since the first three terms of equation (1) form a perfect square, that equation may be written (VTa: + V:B .?/)2 + 2 Cx + 2 F^ + C = . . . (6) wherein the sign of the V5 is the same as that of H. Equation (2) may now be transformed to new axes OX' and 0Y\ which are so chosen that the equation of OX' referred to the given axes shall be ^ VAx+VBy = 0] hence, if 6 be the angle between OX and OX', then tan 6 = ^, whence sin = ~ and cos 6 = — ^:::2z:^r • (J^ VB ' VA +B VA -{- B Equation (7) shows that 6 is negative (if the positive value of y/A + B be used), and acute or obtuse according as V!B is positive or negative. The formulas for transforming to the new axes are (cf . Art. 72) x' -\ y' and y = — ^333:::; x' -\ y . . (0) y/A+B V.4 +B y/A+B -J A + B Substituting these values for x and y in equation (6), it becomes (4 + B)/^ + 2«^^^ZVa^, + 2^^4±^2/' + C = 0. . (9) ^A+B y/A+B 181. J EQUATION OF SECOND DEGREE 305 By dividing equation (9) by (A + B), completing the square of the y'-ierms, and transposing, it may be written in the form _GVZj-FV^) 2 (A+B)^ = -2 GVB - FVA ( ^, ^ {GVA + Fy/B)^ - C(A ^-BY l ^ 2(A +B)^(GVB-FVa) ^ (10) (A + 5)2 Comparing equation (10) with equation [42] (Art. 106), it is seen that the length of the latus rectum, as well as the coordinates of the vertex and focus (with reference to the axes OX' and OY'), and other impor- tant facts, may be read directly from the equation. The advantage of equation (10), over that resulting from the reduction of Ex. 2, Art. 176, is that, in connection with equation (7), it gives all the facts necessary for the immediate location of the curve, and gives those facts in terms of the coefficients of the original equation. Example. Let it be required to determine the position and parameter of the parabola represented by the equation 9x^-2ixy + 16 2/2 -18 a: -101 2/ + 19 = 0. The given equation may be writ- ten as (3 a; - 4 ?/)2 - 18a; - 101 2/ + 19 = 0. If the line Sx — 4:y = 0he chosen as ar'-axis, then tan ^ = f , whence sin^ = — I, and cos 6 = — ^. The formulas of transformation then are : -4a;' + 3^' , 3a;' + 42/' x= ■ — ^ and y= ■ — ^. 5 5 Substituting these values in equa- tion (1), it becomes 25y'^-h70y' = -75x' -19; this equation may be written which shows that the latus rectum is 3, and the coordinates of the vertex and focus (with reference to the new axes) are, respectively, f, — -^ and — ^, — i- It also shows that the axis of the curve is parallel to the negative end of the x'-axis. Recalling the remark about the angle determined by equations (7) above, it is seen that the geometric representation of the above equation is shown in Fig. 123. Fig. 123. TAN. AN. GEOM. 20 306 ANALYTIC GEOMETRY [Ch. XII. 182. Summary. It lias been shown in the preceding articles that every equation of the second degree in two variables represents a conic section, whether the axes are oblique or rectangular ; and that its species and position depend upon the values of the coefficients of the equation. The various criteria of the nature of the conic represented by such an equation, in rectangular coordinates, appear in the following table : The General Equation of the Second Degree Ax^ + 2 Rxi/ + Bi/ -\- 2 ax -\- 2Fy + C = ^ A = ABC + 2 Fas- AF'^ - BG^ - HG^ I. E'^- AB< 0. The ellipse. (1) if A = B, and ir=0, a circle. (2) if A is +, imaginary. (3) if A is — , real. (4) if A is 0, a pair of imaginary straight lines; or, a point. II. B:^-AB = 0. The parabola. (1) if ^ is + , principal axis is the new ^-axis. (2) if ^ is — , principal axis is the new a;-axis. (3) if A is 0, pair of parallel straight lines, which are real and different, real and coincident, or imaginary, according as Cr^ — AC >, = , or < 0. III. R^- AB> 0. The hyperbola. (1) if A=— B^ a rectangular hyperbola. (2) if A is +, principal axis is the new ?/-axis. (3) if A is — , principal axis is the new ic-axis. (4) if A is 0, a pair of real intersecting straight lines. 182-183.] EQUATION OF SECOND DEGREE 307 JSToTE. The above results have not all been shown, but are easily deduced from the work already given. Thus the locus of equation (3), Art. 181, if an ellipse, is imaginary if C" is — ; but, by equation (6), Art. 179, C is — if A is + ; hence the test I (2), given above. And so for the other tests, which the student should verify. The angle which the new axes make with the old, respectively, is chosen as in Art. 175, 2 being taken always positive and not greater than 180°. 183. The equation of a conic through given points. The general equation of a conic may be written Ax'^-^2ffxi/-{-Bf + 2ax-\-2F?/+O=0, . (1) and contains five parameters, the five ratios between the coefficients A^ S^ B^ (x, F^ C. Since five equations, or con- ditions, will determine those parameters, in general five points will determine a conic. That is, in general, a conic may he made to pass through five^ and only five^ given points. If, however, the conic is to be a parabola, one equation is given ; viz. H'^ — AB = 0, hence only four additional con- ditions are needed. In general, a parabola may he made to pass through four points, only. A circle has two conditions given, viz. A = B, 11= 0; therefore, in general, a circle may he made to pass through three points, only. A pair of straight lines has one condition given, A = ; therefore, in general, a pair of straight lines may he made to pass through four points, only. The method to be followed in obtaining the equation of the required conic has been used in Art. 80, and may be indicated for finding the equation of the parabola through four given points. Pi =(2^1, y^), A =(^2^ ^2)' A =(•'^3' ^3)' and P^=(x^. y^. The equation must be of the form (1), 308 ANALYTIC GEOMETRY [Ch. XII. 183. therefore, Ax^^ + 2 Rx^^^ + Bi/^^ + 2 ax^-\-2 Fy^ + C= 0, Ax^^ + 2Hx^^^^-By^ + 2 ax^-\-2Fy^-^ C=0, Ax^ + 2 ^2:3^3 + By^^ + 2ax^ + 2Fy^+C=^, Ax^^2Hx^^-\-By^^ + 2ax^ + 2Fy^-\-O=^0', also, H^-AB = 0. The required ratios between the coefficients of equation (1) may be found from these equations. EXAMPLES ON CHAPTER XII Without transforming the equations to other axes, find the center or the vertex, the axes, and the nature of the following conies : 1. a:2 + 5 a:?/ + 1/2 + 8 a; - 20 ?/ + 15 = ; 2. (x -yy^ + 2x -y = 1] 3. 3x2+2?/2-2a:+.?/-l=0; 4. 3a;2 - 8 a;?/ - 3 .?/2 + a; + 17 ?/ - 10 = 0; 5. 4 a;2 — 4 a:?/ + ?/2 + 4 aa: — 2 a?/ = ; 6. 5a:2 + 2a;2/ + 5/ = 0; 7. 3x2 + 3?/2 + 11 a; - 5^ + 7 = ; 8. a;2 + 2a;y-y2 + 8a; + 4?/-8 = 0; 9. 2/2 — a:?/ — 6 x2 + 1/ — 3 a; = ; 10. y'^ — xy — 5 X + 5 y = 0. Trace the following conies : 11. 3a;2 + 2a:?/ + 3?/2- 16?/ + 23 = 0; 12. 4 a:2 + 9 ?/2 + 8 a; + 36 ?/ + 4 = ; 13. 3 a;2 _ 8 ?/2 + 8 a:?/ - 10 ?/ + 6 a; + 5 = ; 14. (x - y)(x -y -6) +9 =0. 15. What conic is determined by the points (0, 3), (1, 0), (2, 1), (-1,-3), and (3,-3)? 16. Find the equation of the parabola through the points (3, 2), (1, I), (-6, 8), and ("2, |). 17. Find the equation of the conic through the points (9, 2), (6, 3), (3, 2), (1, -2), (2, 1). CHAPTER XIII HIGHER PLANE CURVES 184. Definitions. A curve, in Cartesian coordinates, whose equation is reducible to a finite number of terms, each involv- ing only positive integer powers of the coordinates, is called an algebraic curve ; all other curves are called transcendental curves. Algebraic curves the degree of whose equations exceeds two, and all transcendental curves, are (if they lie wholly in a plane) called higher plane curves. On account of their great historical interest, and because of their frequent use in the Calculus, a few of these curves will be examined in the present chapter. I. ALGEBEAIC CURVES 185. The cissoid of Diodes.* The cissoid may be defined as follows : let OFAK be a fixed circle of radius a, OA a * This curve was invented, by a Greek mathematician named Diodes, for the purpose of solving the celebrated problem of the insertion of tv^o mean proportionals between two given straight lines. The solution of this problem carries with it the solution of the even more famous Delian problem of con- structing a cube whose volume shall be equal to two times the volume of a given cube. For, let a be the edge of the given cube ; construct the two mean proportionals x and y between a and 2 a ; then a : x : -.x :y : :y -.'la, whence x^ = 2- a^, i.e., x is the edge of the required cube. If a = 1, then X = \/2, hence the insertion of two mean proportionals enables one to con- struct a line equal to the cube root of 2. The cissoid may also be employed to construct a line equal to the cube root of any given number (see Klein, Elementargeometrie, S. 35, or the English translation by Professors Beman and Smith). It is not positively known just when Diodes lived ; it is very probable, however, that it was in the last half of the second century b.c. 309 310 ANALYTIC GEOMETRY [Ch. XIII. diameter, AT a, tangent ; draw any line as OQS through 0, meeting the circle in Q and the tangent in S, and on this line lay off the distance OP = QS : the locus of the point P, as the line OS revolves about 0, is the cissoid. * From this definition, the equation of the cissoid, referred to the rectangular axes OX and OT", is readily derived. Let the coordinates of P be a; and ?/, and let O be the center of the circle so that 00= 0A= OK=a. Since triangles 031P and ONQ are similar, ,'.MPi OMi'.NQ: (9iV;.(l) and since OP = QS^ therefore NA = OM = X ; moreover, WQ^= 0]Sr-]VA=(2a-x}x. Substituting these values in equation (1) gives Fig. 124 y \x '. \ V(2 a — x^x : (2 a — x^, whence f = x'- 2a — x^ (2) (3) which is the required rectangular equation of the cissoid. The definition of the cissoid, as well as the equation just derived, shows that the curve is symmetric with regard to * Diodes named his curve ' ' cissoid ' ' (from a Greek word meaning " ivy," because of its resemblance to a vine climbing upwards. The name "cissoid'"' is sometimes, though rarely, applied to other curves which are generated as stated in the definition given above, except that some other basic curve is employed instead of a circle. For other, but equivalent, defini- tions of the cissoid see Note 3, below. 185.] HIGHER PLANE CURVES 311 the a;-axis ; that it lies wholly between the «/-axis and the line x — 2a\ that it passes through the extremities F and K of the diameter perpendicular to OA ; and that it has two infinite branches to each of which the line x = 2a is an asymptote. N'oTE 1. The polar equation of the cissoid referred to the initial line OX, and pole 0, is also easily found. Let the polar coordinates of P be p and 6] then, p = OP = QS = OS - OQ, . . . (4) but OS = 2a sec 6, and OQ = 2 a cos 0, p = 2a sec — 2a cos = 2 a (sec $ — cos 6), i.e., p = 2a tan 6 sin 0, . . . , . (5) which is the polar equation sought. !N"oTE 2. To ''duplicate the cube " by means of the cissoid,* extend CK to H, making HK = CK = a, draw the line HA cutting the cissoid in J, and draw the ordinate EJ. Since CH = 2 CA , therefore EJ = 2 EA ; but from equation (3), ^2 OE^ OE^ _ EA ^EJ' .-. EJ^ = 2 0E\ ... (6) Xow let 111 be the edge of any given cube, and let it be required to construct a line n such that the cube on n shall be equal to the double of the cube on m. Construct n so that OE :EJ:'.m:n', then OE : EJ = m^ : n% and, since EJ^ = 2 . OE^, therefore n^ = 2 m^. XoTE 3. The cissoid may also be defined in either of the following ways : (1) as the locus of the point (P) in which the chord OQS inter- sects that ordinate {ML) of the circle which is equal to iVQ; and (2) as the locus of the foot of the perpendicular let fall from the vertex of a parabola upon a tangent. The derivation of the equation of the curve based upon these definitions is left as an exercise for the student. * To insert two mean proportionals between two given lines by means of the cissoid. See Cantor, Geschichte der Mathematik, Bd. I., S. 339. 312 ANALYTIC GEOMETRY [Ch. XIII. For Newton's method of drawing the cissoid by continuous motion, see Salmon's Higher Plane Curves, p. 183, or Larduer's Algebraic Geometry, p. 196. 186. The conchoid of Nicomedes.* The conchoid may Idc defined as follows : Let PRP' ^ be a -fixed circle of radius a whose center S moves along a fixed straight line OX ; let LK be a straight line drawn through a fixed point A and the center S of this moving circle, and let P and P' be the intersections of this line and the circle ; then the locus traced by P (and by P') as S moves along OX is a conchoid. Y Fig. 125 This definition may also be stated thus : If ^ is a fixed point, OX a fixed line, and S the point in which OX is intersected by a line LK revolving about A^ then the locus of a point P on LK^ so taken that SP is always equal to a given constant a^ is a conchoid. The fixed point A is called the pole, the constant parameter a the modulus, and the fixed line OX the directrix of the conchoid. * The conchoid was invented by a Greek mathematician named Nicomedes, probably in the second century b.c. Like the cissoid, it was invented for the purpose of solving the famous problem of the " duplication of the cube"; it is, however, easily applied to the solution of the related, and no less famous, problem of the trisection of a given angle (see Note 3, below). 186. J HIGHER PLANE CURVES 313 To derive the rectangular equation of the conchoid draw AOY perpendicular, and J.ir parallel, to OX, and let OA=c; let P=(x, y) be any position of the generating point, and draw the ordinate HMP ; then, from the similar triangles ARP and SMP, AH, HP'. iSM.MP, i.e., x: y -{- e '. : Va^ — 'if", y \ [since SM = ^SP - MP =^a^ - ^2], whence a^y =(jj -\- cy^(cfi — ?/2), which is the equation sought. The definition of the conchoid, as well as the equation just derived, shows that the curve is symmetric with regard to the ?/-axis ; that it lies wholly between the two lines y = ci and y = — a\ and that it has four infinite branches to each of which the a;- axis is an asymptote.* Note 1. The polar equation of the conchoid. Let A be the pole, A Y the initial line,' and F = (p, 0) (or P') any position of the generating point; then p = AP = AS ± SP = OA-secO ± SPy i.e., p = c seed ±a, which is the desired equation. Note 2. The conchoid may also be readily constructed by continuous motion as follows : By means of a slot in a ruler, fitting over a pin at A, the motion of the line LK is properly controlled ; if now a guide pin at S, and a tracing point at P, be attached to this ruler, then the point P will trace out the conchoid when the guide point 5 is moved along the line OX. Note 3. By means of a conchoid, any given angle may be trisected.f Let ABC be any angle, on one side (BA) take any distance, as BH, and * It is evident that, ii AO < OB, i.e., H c< a, the curve has an oval below A as shown in Fig. 2 ; if c = a, this oval closes up to a point ; and if c > a, both parts of the curve lie wholly above A. t For the insertion of two mean proportionals between two given lines by means of the conchoid, see Cantor, Geschichte der Mathematik, Bd. I., S. 336. 314 ANALYTIC GEOMETRY [Ch. XIII. draw OHX perpendicular to the other side of the angle (£C) ; then lay- off OK = 2 BH, and construct the conchoid KEF with B as pole and BH = \ OK as modulus, and OX as directrix. Draw HL parallel to BC and counect 5 with L, then the angle LBC = \ABC'j for, join D, the middle point of ML, to H, then ML = OK = 2BH = 2HD, and the three angles marked a are all equal, as are also the two marked (3 ; more- over, f3 = 2a, being the exterior angle of the triangle HLD, which proves that angle LBC = \ABC. 187. The witch of Agnesi.* The witch may be defined as follows : Let OKAQ be a given fixed circle of radius a, OA a diameter, and Q any point on the circle ; if now the ordi- nate MQ be produced to P, so that MQ'.MPiiMA: OA, (1) then the locus of P, as Q moves around the circle, is the witch. To derive the rectangular equation of the witch, let P = (a?, y) be any point on the curve ; then, since MQ ■ = V OM' MA =Vx(2a-x'). * The witch was invented by Donna Maria Gaetana Agnesi (1718-1799) an Italian lady who was appointed professor of mathematics at the University of Bologna, in 1750. 186-188.] HIGHER PLANE CURVES 315 substituting in equation (1) gives ^x(2a- X) -.y ::(2a-x):2a, . . . (2) /=H^^' . • . (3) whicli is the equation sought. The definition of the witch, as well as the equation just derived, shows that the curve is symmetrical with regard to the :?;-axis ; that it lies wholly between the «/-axis and the line x= 2a ', and that it has two infinite branches to each of which the line a; = 2 « is an asymptote. 188. The lemniscate of Bernouilli.* The lemniscate may be defined as follows : let LTARNA' K be a rectangular hyperbola, its center, OX and 01^ its axes, and TE 2i tan- gent to the curve at any point T. Also let 0(r be a perpen- dicular from the center upon this tangent, and let P be the point of their intersection ; then the locus of P as 2^ moves along the hyperbola is called the lemniscate. To derive the rectangular equation of this curve, let OA = a, and let the coordinates of T be x-^^ and y^ ; then the equation of the tangent TU is ^1^ - l/il/ = ^^ • • • (1) hence the equation of 0(7, the perpendicular upon this tan- gent (Art. 62), is x^y -\- y^x = 0. . . . (2) * The lemniscate was invented by Jacques Bernouilli (1654-1705), a noted Swiss mathematician and professor in the University of Basle. It is, how- ever, only a special case of the Cassinian ovals ; viz., of the locus of the ver- tex of a triangle whose base is given in length and position, and the product of whose other two sides is a constant. See Salmon's Higher Plane Curves, p. 44, Gregory's Examples, or Cramer's Introduction to the Analysis of Curves. 316 ANALYTIC GEOMETRY [Ch. XIII. Regarding equations (1) and (2) as simultaneous, the x and 1/ involved are the coordinates of the point P ; more- over, since the point T = (x^,y-^ is on the hyperbola, therefore .2_,/.2^^.2. ... (3) X. Vi = «' Eliminating x-^ and y^ between equations (1), (2), and (3) gives (x^ -\- y'^y = a\a^ - y'^^, . . . (4) which is, therefore, the equation sought. The definition of the lemniscate, as well as the equation just derived, shows that the curve is symmetrical with regard to both coordinate axes ; that it lies wholly between the two lines whose equations are x = — a and x = -{- a \ that it passes through the origin and the two points (— a, 0) and ( + «, 0) ; and that y is never larger than x ; hence the lemniscate is a limited closed curve as represented in Fig. 128. Note 1. The polar equation of the lemniscate is easily derived from equation (4) if the x-axis be chosen as initial line and the origin as pole ; 188.] HIGHER PLANE CURVES 317 for then x = p cos 6 and y = p sin 0, and equation (4) at once reduces to p2 = a2(cos2^-sin2^)=a2cos2^, ... (5) which is therefore the required polar equation of the lemniscate. Equation (5) shows that : when = 0, p =± a; when < 45°, p has two equal but opposite values, each of which is smaller than a ; when = 45°, p = 0, i.e., the angle which the curve makes with the initial line is 45° ; when 45° < ^ < 135°, p is imaginary ; when 135° <0< 180°, p has two equal but opposite values, each of which is smaller than a ; and when 9 — 180°, p =± a. The curve, therefore, consists of two ovals meeting in 0, each lying in the same angle between the asymptotes of the hyperbola as does the corresponding branch of that curve, and these asymptotes are tangent to the lemniscate at the point 0. Note 2. If the two points i^j and F be so located that F^O = OF = — y/2, and if S = (x, y) be any point on the lemniscate, then F^S = ^F^M^ + MS"" = ^j(^ V2 + xY + y% and FS = ^J(^^^/2-xy+y^ hence F^S • FS = ^j(^ V2 + xY-hy^ • -^f | V2'- xY+ y^ = ^|(x^ + y^y-a^ix^-y')+J = |, [by eq. (4)], i.e., F,S •FS = ^. Hence the lemniscate may be defined as the locus of a point which moves so that the product of its distances from two fixed points is con- stant, and equal to the square of half the distance between the fixed points (cf. foot-note, p. 315). This definition of the curve easily leads to the equation already derived ; it also enables one to readily construct the curve thus : with F as center, and any convenient radius FS, describe an arc ; then, with Fj as center, and a third proportional to FS and OF as radius, describe another arc cutting the first in S ; this intersection .S is a point on the locus, and as many points as desired may be constructed in the same way. 318 ANALYTIC GEOMETRY [Ch. XIII. 189^. The limacon of Pascal.* The limagon may be defined as generated from a circle by adding a constant length to each of the radii vectores drawn from a point on its circumference as origin, — proper account being taken of negative radii vectores.f U.g., let OLA^Whe a given — X circle of radius a, any point on it, A^A = k any constant ; then if any radius vector as OP^ be drawn from 0, and PjP = A^A = k he added to I it, then P is a point on the limagon ; and as P^ is made to describe a circle, P will trace the limagon. The polar equation of the curve is at once written down from this definition ; for, if the diameter OCX be taken as initial line, then the polar equation of the circle is p = 2 a cos 9, . . . (1) whence the polar equation of the limagon is p = 2acQse -\-k. . . . (2) If k be taken equal to a, the radius of the given circle, this equation may be written in the more common form p = a(l + 2cos0}. ... (3) * This curve was invented and named by Blaise Pascal (1623-1662), a celebrated French geometrician and philosopher. It is, however, a special case of the so-called Cartesian ovals. t The limacon may also be defined as the locus of the intersection of the two lines OP and CP which are so related during their revolution about and C, respectively, that the angle XCP is always equal to | times the angle XOP. This definition easily leads to the polar equation already derived. 189«-1896.] HIGHER PLANE CURVES 319 The definition of the limagon, as well as the equation just derived, shows that the curve is symmetrical with regard to the initial line, and that it has the form shown in Fig. 129. Note. The rectangular equation of the lima9on for which ^ = a is easily derived from equation (3). Choosing the initial line and a perpen- dicular to it through as rectangular axes, so that x = p cos 0, and y = psin 6, equation (3) becomes Vx-2 + //^ = a + 2 g . ... (4) y/x"^ + y'^ Rationalizing equation (4) gives (x2 + f -2 axy = a%x^ + tf), . ... (5) which is the usual form for the rectangular equation of the lima9on. 189^. The cardioid. The cardioid may be defined as a special case of the limagon ; viz., it is a limagon in which the constant k, which is added to each of the radii vectores, is taken equal to the diameter of the fundamental circle. If in the equation of the limagon [Art. 189a, equation (2)] the constant k be taken equal to 2 a, that equation becomes p = 2a(l + cos 6>), . . . (1) which is the polar equation of the cardioid. The more usual form in which the equation of the cardioid is written is /3 = 2a(l — cos^), ... (2) but this amounts merely to turning the figure through 180° in its own plane. Note 1. The rectangular equation of the cardioid is obtained as in Art. 189 a. p^ » It is (x^ + ?/2 + 2 axy = a%x^ + y^) . (3) /%\ ^ The curve represented by equations (2) f/^ \/\ and (3) has the form shown in Fig. 130. -I Mi The cardioid is usually defined as the vv_^^ \. locus traced by a point on a given circle \^ ^ AKAyL, which rolls on an equal but fixed >v circle OMA.H. This definition also leads to ' Fig 130 equations (2) and (3) already derived. 320 ANALYTIC GEOMETRY [Ch. XIII. 190. The Neilian, or semi-cubical, parabola.* This curve may be defined as follows : let HTASKL be a given parab- ola whose equation is let TMS be any double ordinate of the curve, TT^ a tangent at the point -X T=(x^^ ?/j), and AQ d^. perpendicular from the vertex upon this tangent ; if QA intersects TS in P, then the locus of P as 2^ moves along the parabola is called a semi-cubical or Neilian parabola. Its rectangular equation is derived as follows : the equa- tion of r2\is y^y = 1p(x + x;), . . . (2) hence the equation of ^^ is Fig. 131 y=-fy- (3) The equation of TS is JO — Ju-t» • • • I "x J If now equations (3) and (4) be regarded as simultaneous, then X and y are the coordinates of the point P in which the two lines intersect, and if x-^ and y^ be eliminated by means of the equation y^==4,px-^, ... (5) an equation connecting x and y is obtained. * This curve is historically interesting, because it is the first one which was rectified^ i.e., it is the first one the length of an arc of which was expressed in rectilinear units. This celebrated rectification was performed, without the aid of the modern Calculus methods, by William Neil, a pupil of Wallis (see Cantor, Geschichte der Mathematik, Bd. IL, S. 827), in 1657, and is, for that reason, called the Neilian parabola. It is also called the semi- 3 cubical parabola because its equation may be written in the form y = ax^. 190-191.] HIGHER PLANE CURVES 321 Substituting for x^ and y^, in equation (5), their values in terms of x and ^ as found from equations (3) and (4), gives t.e.<, 4^2 f: = i2 f (6) which is the equation sought. This equation shows that the curve passes through the origin and is symmetrical with regard to the a:;-axis ; that it lies wholly on the same side of the y-axis as does the given parabola ; and that it has two infinite branches. II. TRANSCENDENTAL CURVES* 191. The cycloid. t The cycloid (^OPKA) is the path traced by a point P on the circumference of a circle {HNSP^ Y * A few very common transcendental curves have already been examined in Chapter III ; among these are the curve of sines, the curve of tangents, and the logarithmic curve. t Because of the elegance of its properties, and because of its numerous applications in mechanics, the cycloid is the most important of the transcen- dental curves. It has the added historical interest of being the second curve that was rectified (cf. Art, 190, foot-note). Its rectification was first accom- plished by Sir Christopher Wren (1632-1723) and published by him in 1673. TAN. AN. GEOM. 21 322 ANALYTIC GEOMETBY [Ch. Xm. which rolls, without sliding, upon a fixed right line (OX). The point P is called the generating point ; the circle PHNS^ the generating circle ; the points and A^ the vertices ; the line EK, perpendicular to OA at its middle point, the axis ; and the line OA^ the base of the cycloid. To derive the rectangular equation of the cycloid let a be the radius of the generating circle, and OX the fixed straight line on which it rolls ; also let P be the generating point, and let PNS be any position of the generating circle. Draw the radius CP^ the ordinate MP^ the line PL parallel to OJl, and the radius OH to the point of contact of the generating circle and the line OX. Let OX and OY (the perpendicular to it through 0) be chosen as axes, and let e be the angle PCH. Then, if P =(x., ?/), x= 0M= OH-MH = OH -PL = aO — a sin ^, [since 0H= 'drGPH= a6^. i.e., X — a(^0 — sin 0^. . . . (1) Similarly, q/ = a(l — cos ^). . . . (2) Solving equation (2) for d gives /I a — y cos 6 = ^, a i.e., 6 = cos-^ ( ^ ) = vers-^ ( ^ ) ; \ a J \aj and substituting this value of in equation (1) gives x = a vers~^ ( ) ~" V2 ay — y'^, . . . (3) which is the rectangular equation sought. 191-192.] HIGHER PLANE CURVES 323 Note 1. It is usually simpler to regard equations (1) and (2) together as representing the cycloid ; 6 is then the independent variable, while x and y are both functions of it. Note 2. The cycloid belongs to the kind of curves called roulettes. These curves are generated by a point which is invariably connected with a curve which rolls, without sliding, upon a given fixed curve. If both the rolling and the fixed curves are circles, then the curve generated is designated by the general name of trochoid. If the gen- erating point is on the circumference of the rolling circle, and this circle rolls on the outside of a fixed circle, then the curve described is called an epicycloid ; but if it rolls on the inside of the fixed circle, the generated curve is called a hypocycloid. The cycloid may be regarded either as an epicycloid or a hypocycloid, for which the fixed circle has its center at infinity and an infinite radius. 192. The hypocycloid. Let the hypocycloid APRST •• be traced by the point P on the circumference of the circle PQR, whose radius is 6, and which rolls on the inside of the Y s FigI 133 fixed circle AQE, whose radius is a. Also let P= (x, y) be any position of the generating point. Draw the line OOQ, the ordinates EO and MP, the radius OP, and the 324 ANALYTIC GEOMETRY [Ch. Xlll. line KF parallel to OA, where A is the point with which P coincided when in its initial positiono Let OAX and OY, the perpendicular to it through 0, be chosen as coordinate axes; also let the angles AOQ, FO'Q and O'FK be desig- nated, respectively, by 6, 6' and <^. Then 0M= OH + EM= OH -\- KF = O0'cosl9 + P0'cos(^ = Oa cos e + FO' COS {6' - 6>), [since <^ = S' - 6^ i.e., x=(a- h) cos6 + b cos (6' _ 6>). . . . (1) But since arc AQ = sltc FQ, therefore a0 = b6', whence $' =-^, and equation (1) becomes b x = (a-b)cose + b cos ^^ ~ ^^ ^ . ... (2) Similarly, ^ =(^a — b') sinO — b sin^^ — ^ . . . . (3) Equations (2) and (3) are together the equations of the hypocycloid. A single equation representing the same curve may be found, as in the case of the cycloid (Art. 191), by eliminating 6 between equations (2) and (3). Note. If the radii of the circles be commensurable, i.e., if b equals a fractional part of a, then the hypocycloid will be a closed curve ; but if these radii are incommensurable, then the curve will not again pass through the initial point A. In particular, if 0:6 = 4:1, then the circumference of the fixed circle is 4 times that of the rolling circle, and the hypocycloid becomes a closed curve of four arches, as shown in Fig. 134. In this case, equations (2) and (3) become, respectively, 192-194.] HIGHER PLANE CURVES 325 2 X = ^a cos ^ + ^ a cos 3 ^, and y = ^a^mS — \a sin 3 0. But, by trigonometry, 3cos^+ cos 3^=i4cos3^, and 3 sin — sin 3^ = 4 sin^ 0, hence equations (4) become X = a cos^ 0, and y = «sin^ 0; 2. 2l whence x^ -\- y^ which is the common form of the equation of the four-cusped hypocy- cloid. SPIRALS 193. A spiral is a transcendental curve traced by a point which., while it revolves about a fixed point called the center, also continually recedes from this center, according to some definite law. The portion of the spiral generated during one revolution of the tracing point is called a spire ; and the circle whose radius is the radius vector of the generating point at the end of the first revolution is called the measuring circle of the spiral. Thus, in Fig. 135, ABODE is the measuring circle, OQSUWA is the first spire, and J.l^_ZrXiV is the sec- ond spire. 194. The spiral of Archimedes. f This curve is traced by a point which moves about a fixed point in a plane in such a * If this equation be rationalized, it becomes 27 a2xV ^ (^: A — w2>3 2/2)3. Although the hypocycloid is, in general, a transcendental curve, it becomes algebraic for particular values of the ratio of the radii of the circles. t This curve is usually supposed to have been discovered by Conan, though its principal properties were investigated by the geometer whose name it bears. 326 ANALYTIC GEOMETRY [Ch. XIII. way that any two radii vectores are in the same ratio as are the angles they make with the initial line.* From this definition it follows that the equation of the curve is p = ke, . . . (1) where A; is a constant. This equation shows that the locus passes through the origin, and that the radius vector becomes larger and larger without limit as the num- ber of revolutions increases without limit. Moreover, if (.Pv ^i) ^® ^^y point on the curve, and if (/O2, ^^ + 2 tt) be the corresponding point on the next spire, then p^ = hO-^ and p^ — k(6^ + 2 tt), whence p^ = p^-\- '2k7r\ but 2k7r = OA^ hence the distance between the successive points in which any radius vector meets the curve is constant ; it is always equal to the radius of the measuring circle. This follows also directly from the definition. The locus of equation (1), for positive values of 6 is rep- resented in Fig. 135 ; for negative values of 6 the locus is symmetrical with the part already drawn, the axis of sym- metry being the line LF. 195. The reciprocal or hyperbolic spiral. This curve is traced by a point which moves about a fixed point in a plane in such a way that any two radii vectores are in the * This curve may also be defined thus : It is the path traced by a point which moves av^ay from the center with uniform linear velocity, while its radius vector revolves about the center with uniform angular velocity. 194-195.] HIGHER PLANE CURVES 327 same ratio as the reciprocals of the angles which they form with the initial line. From this definition it follows that the equation of the curve is p = -, . . . (1) where ^ is a constant. This equation shows that the curve begins at infinity when ^ = and winds round and round the center, always approaching it, but never quite reaching it ; i.e., p = only after an infinite number of spires have been described. Equation (1) also shows that the constant k is the cir- cumference of the measuring circle. For the radius of the measuring circle (Art. 193) is the radius vector of the gener- ating point of the curve at the end of the first revolution, i.e., when ^ = 2 tt ; but, from equation (1), this radius vector is — ■, and the circumference of the circle of which this is 27r the radius is k. Again, if P = (/?, ^) be any point on the locus of equa- tion (1), then pe = k = circumference of measuring circle ; but pO equals the length of the circu- lar arc described with radius p and subtending an angle 0, therefore the length of any circular arc as MP, described about 0, with radius p, and extending from the initial line to the curve, is equal to the circum- ference of the measuring circle. The locus of equation (1), for positive values of 6, is represented in Fig. 136. 328 ANALYTIC GEOMETRY [Ch. XIII. 196. The paracolic spiral. This curve is traced by a point which moves around a fixed point in a plane in such a way that the squares of any two radii vectores are in the same ratio as are the angles which they form with the initial line. From this definition it follows that the equation of the •^•^'■^'^^ ^ = ke, . . . (1) where A; is a constant. This equation shows that the curve begins at the center when ^ = 0, winds round and round this point, always receding from it, the radius vector becoming infinite when 6 becomes infinite, ^.e., when -R it has described an infinite number of spires. The locus of equation (1), for positive values of /o, is represented Fig. 137 ^j^ ^ig. 137.* 197. The lituus f or trumpet. This curve is traced by a point which moves around a fixed point in a plane in such a way that the squares of any two radii vectores are in the same ratio as the reciprocals of the angles which they form with the initial line. From this definition it follows that the equation of the curve is P^ — a^ • • • C^) u where A; is a constant. This equation shows that the curve begins at infinity, when ^ = 0, and winds round and round the center, always * See also Eice and Johnson's Differential Calculus, p. 307. t This curve was invented and named by Cotes, who died in 1716. 196-198.] HIGHER PLANE CURVES 329 approaching it, but never quite reaching it, i.e.^ f> = only after an infinite number of spires have been described. The locus of equation (1) is shown in Fig. 138 ; the heavy Fig. 138 line being the part of the locus obtained from the positive values of p, while the dotted part belongs to the negative values of p. Note. The four spirals just discussed, and whose forms are given in Figs. 135 to 138, are all included under the more general case of the curve defined by the equation _ ^^ if n = 1, this is the spiral of Archimedes; if n = — 1, it is the hyperbolic spiral ; if n = |, it is the parabolic spiral ; while if n = — I, it is the lituus. 198. The logarithmic spiral.* This curve is traced by a point which moves around a fixed point in a plane in such * This curve might have been defined by saying that the radius vector increases in a geometric ratio while the vectorial angle increases in an arith- metic ratio. An important property of this curve is (see McMahon and Snyder's Differential Calculus, Art. 120) that it cuts all the radii vectores at the same angle, and the tangent of this angle is the modulus of the system of logarithms which the particular spiral represents. 330 ANALYTIC GEOMETRY [Ch. XUI. 198. a way that the logarithms of any two radii vectores are in the same ratio as are the angles Avhich these lines form with the initial line. From this definition it follows that the equation of the curve is log p = kd^ . . . (1) where Jc is a constant. If k be unity, and logarithms to the base a be employed^ this equation may be written in the form p = a'. . . . (2) This equation shows that if ^ = — oo, p= ; that p in- creases from to 1, while 6 increases from — go to ; and that p continues to increase from 1 to go, while increases from to + GO ; the curve has, therefore, an infinite number of spires. If the constant a equals 2, then p takes the values •••■!, |-, 1, 2, 4, 8, •••, when 6 is assigned the values (in radians), ...^ _ 2, — 1, 0, 1, 2, 3, ••• ; Fig. 139 represents the locus of equation (2), a being equal to 2, for values of 6 from — 2 tt to +3. In this figure Z FOE = Z EOA =Z AOB =Z BOO = ZO0B=bT.%, and OF = \, OE = ^, 0A=1, OB = 2, 0(7=4, and OB = S. Fig. 139 PART II SOLID ANALYTIC aLOMETRY CHAPTER I COORDINATE SYSTEMS. THE POINT 199. Solid Analytic Geometry treats by analytic methods problems which concern figures in space, and therefore in- volve three dimensions. It is evident that new systems of coordinates must be chosen, involving three variables ; and that the analytic work will therefore be somewhat longer than in the plane geometry. On the other hand, since a plane may be considered as a special case of a solid where one dimension has the particular value zero, it is to be expected that the analytic work with three coordinate vari- ables should be entirely consistent with that for two vari- ables ; merely a simple extension of the latter. The student should not fail to notice this close analogy in all cases. In the present chapter will be considered some simple and useful sj^stems of coordinates for determining the position of a point in space, some elementary problems concerning points, and the transformations of coordinates from one system to another. Later chapters Avill treat briefly of surfaces, par- ticularly of planes and of surfaces of the second order, and of the straight line. 331 332 ANALYTIC GEOMETRY [Ch. I. Z N X- ,Y' O N ^M ^X M' Fig. 140 200. Rectangular coordinates. Let three planes be given fixed in space and perpendicular to each other, — the coordi- nate planes XOY^ YOZ^ and ZOX, They will intersect by pairs in three lines, X' X^ Y'Y, and Z'Z, also perpen- dicular to each other, called the coordinate axes. And these three lines will meet in a common point 0, called the origin. Any three other planes, MF, AT, and LF, parallel respectively to these coordinate planes, will intersect in three lines, A^'P, L'P^ M'P^ which will be parallel respectively to the axes ; and these three lines will meet in, and completely determine, a point P in space. The directed distances A'P, X'P, and M'P thus determined, i.e.^ the perpendicular distances of the point P from the coordinate planes, are the rectangular coordinates of the point P. They are represented respec- tively by ic, ?/, and z. It is clear that X == N'P = LL' = NM' = OM; y = HP = MM' = LN' = ON-, z = M'P = AAT' = ML' = OL. It is generally convenient, however, to consider x= OM, y = MM', and z = M'P. The point may be denoted by the symbol P = (^x, y, z). The axes may be directed at pleasure ; it is usual to take the positive directions as shown in the figure. Then the eight portions, or octants, into which space is divided by the coordinate planes, will be distinguished completely by the signs of the coordinates of points within them. 200-202.] THE POINT IN SPACE 333 If the chosen coordinate planes were oblique to each other, a set of oblique coordinates for any point in space miglit be found in an entirely analogous way. Unless otherwise stated, rectangular coordinates will be used in the subsequent work. 201. Polar coordinates. A second method of lixing the position of a point in space is by means of its distance and direction from a given fixed point. Let ^ be a fixed point in space, called the pole ; and let p be the distance from to any other point P. To give the direction of p, let OH and OS be two chosen directed perpendicular lines through 0, determining the plane ROS; then the direction of p will be Fig. ui given by the angle 6 from the plane BOS to the plane POJ/, and the angle (f) from the line OS to p. The point P is completely determined by the values of its radius vector p and its vectorial angles and <^, and may be denoted as P = (p, 6, (/)). The elements p, 6, cj) are called the polar coordinates of the point P. It is to be noted that for convenience the positive values of 6 and ^ are those for rotation in clockiuise direction from ROS and OS^ respectively. And although a given set of coordinates fixes a single point, yet any point may have eight sets of coordinates in a polar system, if, as usual, the valuer of the angles are less than 360°. 202. Relation between the rectangular and polar systems. If the axes OR and OS of a polar system coincide with the axes OX and OZ, respectively, of a rectangular sys- 3M ANALYTIC GEOMETRY [Ch. I. and that is, Again, ^.e., also tern, the pole and origin therefore being coincident, then simple rela- tions exist between the two sets of coordinates for any point. For, since Z OMM' = 90° and Z OM'F= 90°, therefore OM ^ OM' cone = OP sin cos 0. MM' = OM' sin = OF sin sin 6, M'F = OP cos (^ ; a? = pcos0sm<|), 1 2/ = P sin e sin <}), f . • . [1] s = pcos4>. J OF^ = OW^ + WF^ = OM^ + MM^ + M^. p2=:Cc2 + |/2 + »2, tane=^, a? and cos = ^^a?'^ + y^ -i-z^. . [2] The above relations give formulas for transformation from the one coordinate system to the other. 203. Direction angles : direction cosines. A third useful method of fixing a point in space is a combination of the two methods already considered. The axes of reference are chosen as in rectangular coordinates, and any point F of space is fixed by its distance from the origin, called the radius vector, and the angles a, /3, 7, which this radius Fig, 143 202-203.] THE POINT IN SPACE 335 vector makes with the coordinate axes, respectively. These angles are called the direction angles of the line OP, and their cosines, its direction cosines. The point may be con- cisely denoted as the point P = (/o, a, /3, 7) . Simple equations connect these coordinates with those of the rectangular system ; for, projecting OP upon the axes OX, OY, and OZ, respectively, a? = pCOSa, t/ = pCOSp, » = pCOSYj . . . [3] and also, p^ = x^ -{- y'^ -\- z^, as in equations [2]. Moreover, the direction cosines are not independent, but are connected by an equation ; for, by combining the above equations, ^ = p2 qqq2 a-i- p^ cos^ /3 + p^ COS^ y^ i.e., cos'^a + cos^|3 + cos''^7 = 1. . . . [4] Such a relation was to have been expected, since only three magnitudes are necessary to determine the position of a point, and therefore the four numbers /o, «, yS, 7 could not be independent. Any three numbers, a, b, c, are proportional to the direc- tion cosines of some line ; because if these numbers are con- sidered as the coordinates of a point, then the direction cosines of the radius vector of that point are, by eq. [3], cos a= — ^ , COS 3= ^ , COS 7= . [51 Va2 + ft2 + c2 Va2 + 52 + c2 Va2^_2,2_,.c2 These direction cosines are proportional to a, b, c; and are found by dividing a, b, c, respectively, by the same constant, Va2 + 52 + c2. Direction cosines are useful in giving the direction of any line in space. The direction of any line is the same as that of a parallel line through the origin, therefore the direc- tion of a line may be given by the direction angles of some 886 ANALYTIC GEOMETRY [Ch. I. point wliose radius vector is parallel to the line. Sometimes, as an equivalent conception, it is convenient to consider the direction angles as those formed by the line with three lines which pass through some point of the given line, and are parallel, respectively, to the coordinate axes. 204. Distance and direction from one point to another ; rec- tangular coordinates. A few elementary problems concerning points can now be easily solved ; for example, the problem of find- ing the distance between two points. Let OX, OY, OZ be a set of rectangular axes, and be two given points. Then the planes through P^ and P^^ paral- lel, respectively, to the coordi- nate planes, form a rectangular parallelopiped, of which the required distance P^P^ is a diagonal. From the figure, since Z P^QP^ = 90° and Z M^RM^ = 90°, therefore P^ = I^Q" + W? = WW + ~Q^i^ = MlR^ -f- BM^ -{- QP^ = (^2 - ^l)^ + (^2 - ^l)^ + C^2 - ^lY' That is, if d be the required distance, Fig. 141 d= ^(0^2 - a?i)^ + (2/2 - 2/i)- + ^z,2- zi)^ [6] Moreover, since the direction of the line PjPg ^^ given by the angles a, /3, 7, which it makes, respectively, with the lines P^X^ ^lY'^ and P^Z', drawn through Pj parallel to the 203-205. THE POINT IN SPACE 337 axes ; then projection oi d = P^P^ upon these lines in turn gives P^F^GOsa=P^X', PlP2Cos^=Pl^^ F^P^cosy=P^Z\ ^.e., c?cos a = 2-2 — a?i, dcos ^ = ^2~ 1/v c? cos 7=^2 — ^i? and, finally, COSa= OCi} — OC-t cosp y^-Vi d ' —- d ' «««Y = — ^— • • • L^J Tlifise equations give the required direction angles of 20s The point which divides in a given ratio the straight line from one point to another. Let z be two given points, and let Pg = (a?3, ^/g, ^3) be a third point which divides the line P1P2 in the given ratio — i, so that ^^ = — 1. m^ P^P^ ^^ m; 2 F/ Fig. 145 M, Let P1P3 = c?i, and P3P2 = d^ ; then by Art. 204, if «, /3, 7 be the direction angles of P^P^-, •^1 **/0 ♦t/O *t/Q **^1 COS Ct = -^ d-i 3. l_ ^ c?. ^2 *^3 C?r and Similarly, and > i»3 = wiia?2 + iW2a?i yo _ 9 m^z.2 + in.yZi Zo = — . nil + ^^2 [8] It will be noticed, as in the similar problem in Part I, Art. 30, that if P3 divides the line externally, the ratio — 1 ???2 must be negative ; and the above formulas still apply. TAN. AN. GEOM. — 22 338 ANALYTIC GEOMETRY [Ch. I. If Pg bisects the line P^P^^ formulas [8] take the simpler forms ^3- 2 ' ^^~ 2" — ' ^^ — 2 — * • • • L^J But 206. Angle between two radii vector es. Angle between two lines. Let P^ = (p^, «^, ^^, 7^) and P^ = (p^, a^, P^, 73) ^® two given points, and 6 the angle included by the radii vec- tores p-^ and p^. Then the pro- jections upon OP^ of the line OP^ and of the broken line OMJSI^P^ are equal (Art. 17); hence, proj. OP2 = proj. OM^M^P^, i.e., P2 cos =0M2 cos'^i + M^M^' cos ySi + M^'P^ cos 7i. OM2 = P2 cos «2i JfgMg' = P2 COS 72, and ifg' A = /'2 ^^s 72 ; hence, P2 COS ^=/32 COS ttg cos «i + /02 COS ySg COS ^i-\-p2 COS 73 COS 7j *, ie., cos 8 = cos ai COS a2 + COS Pi COS P2 + COS 71 COS 72> [1^] and this relation determines the required angle 0. It follows, since any two straight lines in space have their directions given by the direction angles of radii vectores which are parallel to them, respectively, that formula [10] applies as well to the angle 6 between any two straight lines in space, whose direction angles are given. Two special cases arise, of parallel and of perpendicular lines. If the two given lines are parallel, evidently ^^1 = ^2' Pi = P2' vi = 72; [11] 205-207.] THE POINT IN SPACE 389 and formula [10] reduces to eq. [4]. If the lines are per- pendicular, cos ^ = 0, and eq. [10] reduces to cos aj COS a2 -f COS Pi COS Pg + COS Vi COS 72 = 0. . . . [12] 207. Transformation of coordinates; rectangular systems. The relations found in Art. 202 to exist between rectangu- lar and polar coordinates of a point may be used as formulas of transformation from one system to the other if the origin, the pole, and the reference axes are coincident. Two other simple transformations may be useful, (1) from one set of rectangular coordinates to a parallel set, i.e., a change of origin only ; and (2) from one set of rectangular axes to another set through the same origin, i.e., a change of direc- tion of axes. Then any transformation between rectangular and polar systems can be per- formed by a combination of these three elementary trans- formations. (1) Change of origin only. Let the new origin be the point 0' = (^h, k, y); then, construct- ing the coordinates of any point P with reference to each set of coordinate planes, it is evident, by analogy with Art. 71, that Fig. 147 oi> = a>' + h, y-y' + h, z = z'+J, [13] (2) Change of direction of axes. Let a second set of rec- tangular axes, OX', 0Y\ OZ', have the direction angles «,, ^v Tr ^T Pv yv ''^^^^^ H-' ^B^ 73' respectively, with the old axes OX, OY, OZ, 340 AJ^ALTTIC GEOMETRY [Ch. I. [14] Then if the coordinates of any point P in the two systems are X = OM, y = MM\ z = 31' P, and x' = OQ, y = QQ'. z' = Q'P. then projections of OP and the broken line OQQ'P upon OX, OY^ OZ^ in turn, will be equal; hence, 00 = oc' COS aj + y' COS a^ + z' COS a3, 1/ = a?' COS Pi + 2/' COS po + z' cos P35 z = x' cos 7i + y' cos 72 + ^' COS 73. These formulas are for transformation from the first sys- tem to the second. But, also, by projecting OP and OMM'P upon 0X\ 0Y\ 0Z\ respectively, x' = X COS «i + ?/ COS (B^-\- z cos 7j, " y' =z X cos «i + ^ cos 13^ + Z COS 721 ' • • • [15] z' = X COS «3 + y COS ^83 + 2 COS 73, and these formulas are for the reverse transformation, from the second system to the first. Note. It is to be remembered that in the transformation of [14] and [15], twelve conditions exist, by eq. [4] and eq. [12], three of each of the following types, cos^ttj + cos^ag + cos'^ag = 2, cos^a^ + cos2/?j+ cos'^y^ = 1, cos ttj^ cos a.2 + cos (i^ cos ^^ + ^^^ Yi ^os yg = 0, cos a^ cos ^j + cos a^ cos ^^ + ^^s a.^ cos /Jg = 0. These equations are not independent, however, but reduce to six independent equations. 207]. THE POINT IN SPACE 341 It is clear, by reasoning similar to that of Art. 75, Part I, that none of the transformations [13], [14], and [15], neither separately nor in combination, can alter the degree of an equation to which they may be applied. EXAMPLES ON CHAPTER I 1. Prove that the triangle formed by joining the points (1, 2, 3), (2, 3, 1), and (3, 1, 2), in pairs, is equilateral. 2. The direction cosines of a straight line are proportional to 1, 2, 3 ; find their values. 3. Find the angle between two straight lines whose direction cosines are proportional to 2, 2, 2, and 5, ~4, 7, respectively. 4. The rectangular coordinates of a point are (V3, 1, 2V3); find its polar coordinates. 5. The polar coordinates of a point are (8, -, ^j; find its rectan- gular coordinates. 6. Express the distance between two points in terms of their polar coordinates. 7. Find the coordinates of the points dividmg the line from (~2, -3, 1) to (3, ~2, 4) externall}^ and internally in the ratio 2 : 5. 8. What is the length of a line whose projections on the coordinate axes are 4, 1, 3, respectively? 9. Find the radius vector, and its direction cosines, for each of the points (-7, 1, 5), (1, -1, -2), (a, 0, h). lOo Find the center of gravity* of the triangle of Ex. 1. 11. Find the direction angles of a straight line w^hich makes equal angles with the three coordinate axes. 12. A straight line makes the angle 30° with the rc-axis, and 75° with the 2-axis. At what angle does it meet the ?/-axis ? 13. Prove analytically that the straight lines joining the mid-points of the opposite edges of a tetrahedron pass through a common point, and are bisected by it. 14. Prove analytically that the straight lines joining the mid-points of the opposite sides of any quadrilateral pass through a common point, and are bisected by it. * See Ex. 15, p. 42. CHAPTER II THE LOCUS OF AN EQUATION. SURFACES 208. Attention lias been called to tlie close analogy between the corresponding analytical results for the geom- etry of the plane and of space. It is evident that in geometry of one dimension, restricted to a line, the point is the elementary conception. Position is given by one vari- able, referring to a fixed point in that line ; and any alge- braic equation in that variable represents one or more points. In geometry of two dimensions, however, it has been shown that the line may be taken as the fundamental element. Position is given by two variables, referring to two fixed lines * in the plane ; and any algebraic equation in the two variables represents a curve, z.e., a line whose generating point moves so as to satisfy some condition or law. Corre- spondingly, in geoDietry of three dimensions the surface is the elementary conception. Position is given by three variables, referring to three fixed surfaces, since any point is the inter- section of three surfaces ; f and it can be shown that any algebraic equation in three variables represents some surface. * Witli polar coordinates, these lines are a circle about the pole with radius = p, and a straight line through the pole making the angle 6 with the initial line (Art. 23). t With polar coordinates, these surfaces are a sphere, about the origin as center, determined by the radius vector p, a right cone about the ^-axis, with vertex at the origin, determined by the angle ^, and a plane through the s-axis determined by the angle d (Art. 201). 342 208-209.] SUBFACES 343 The study of the special equations of first and second degree will be taken up in the two succeeding chapters. Here it is desired to show that an algebraic equation in three variables represents a surface, and to consider briefly two simple classes of surfaces : (1) cylinders, ^.6., surfaces which are generated by a straight line moving parallel to a fixed straight line, and always intersecting a fixed curve ; and (2) surfaces of revolution, z.e., surfaces generated by revolving some plane curve about a fixed straight line lying in its plane. 209. Equations in one variable. Planes parallel to coordi- nate planes. From the definition of rectangular coordinates, it follows that the equations a; = 0, ^ = 0, 2 = 0, represent the coordinate planes, respectively, and that any algebraic equation in one variable and of the first degree represents a plane parallel to one of them. Similarly, an equation in one variable and of degree n will represent n such parallel planes, either real or imaginary. For, any such equation, as can be factored into 7i linear factors, real or imaginary, FoC^-^i)(^-^2)(--0(^-^J= 0; ... (2) and by the reasoning of Part I, Art. 40, eq. (2) will repre- sent the loci of the n equations X — x-^ = 0, X — X2 = 0^ ••., X — x„ — 0, each of which is a plane, parallel to the ^^-plane, and real if the corresponding root is real. In the same way, an equa- 344 ANALYTIC GEOMETRY [Ch. II. tion in y or 2 only will represent planes parallel to the zx- or Any algebraic equation in one variable represents one or more planes parallel to a coordinate plane. It follows at once by Art. 39, that two simultaneous equations of the first degree in one variable represent the intersection of two planes parallel to coordinate planes ; therefore, represent a straight line parallel to the coordi- nate axis of the third variable ; e.g.., y = 1., z = c, considered as simultaneous equations, represent a straight line parallel to the iT-axis. 210. Equations in two variables. Cylinders perpendicular to coordinate planes. Consider the equation 2a; + 3^ =6, (1) with two variables only. In the a:?/-plane it represents a straight line AB. If, now, from any point F of AB a Fig. 149. straight line be drawn parallel to the 2-axis, the x and y coordinates of every point Q on this line will be the same as for P, and therefore satisfy equation (1). Moreover, if the line FQ moved along AB^ and always parallel to the 2-axis, 209-210.] SURFACES 345 still the coordinates of every point in it satisfy equation (1) . As the line P^ is thus moved, it traces a plane surface per- pendicular to the o^^-plane ; and, as evidently the coordinates of a point not on this surface do not satisfy equation (1) this cylindrical plane is the locus of equation (1). Again: the equation f^z'^=r^ . . . (2) represents in the ^^-plane a circle. It is therefore satisfied by the coordinates of any point $, in a line parallel to the ^-axis, through any point P of this circle ; and also by the coordinates of Q as this line PQ is moved, parallel to Fig. 150, the X-axis and along the circle. The circular cylinder thus traced by the line PQ, perpendicular to the ^^-plane, is the locus of the given equation. Similarly, it may be shown that the locus of the equation ^_ ^_ -J 52 (3) is a cylindrical surface traced by a straight line parallel to the y-axis, and moving along the hyperbola whose equation in the a;2-plane is equation (3). And, in general, it is clear by analogy that an^ algebraic equation in two variables repre- sents a cylindrical surface whose elements are parallel to the 346 ANALYTIC GEOMETRY [Ch. n. axis of the third variable^ and having its form and posi- tion determmed hy the plane curve represented by the same equation. As a direct consequence, it is clear that if a cylinder lias its axis parallel to a coordinate axis, a section made by a plane, perpendicular to that axis, is a curve parallel to and equal to the directing curve on the coordinate plane, and is represented in the cutting plane by the same equation. Thus, the section of the elliptical cylinder whose equation is 3 x^ -\- y'^ = 5, cut by the plane ;3 = 7, is an ellipse equal and parallel to the ellipse whose equation is 3 a;^ + ^^ = 5. 211. Equations in three variables. Surfaces. A solid figure has the distinctive property that it can be cut by a straight line in an infinite number of points, while a sur- face or line can, in general, be cut in only a finite number. A line has the distinctive property that it can be, in gen- eral, cut by a plane in only one point, while a surface may be cut in a curve. To show that the locus of an algebraic equation in three variables is, in general, a surface, it is suf- ficient to show that, in general, a plane will cut it in a curve, while a straight line will cut it in a finite number of points. Let the given equation be f(x,y,z)=^, . . . (1) and let z = c . , . (2) be a plane parallel to the a^^-plane. The points of inter section of these two loci will be on the locus of the equation f(x, y, c)=0; . . . (3) and, by Art. 210, they lie, therefore, upon a plane curve, cut from the cylinder whose equation is (3), by the plane whose equation is (2). Hence the locus of equation (1) is not a line. 210-212.] - SURFACES 347 Again, let ^ z= h, z = o . . . (4) be the equations of a straight line (Art. 209), parallel to the a;-axis. The points of intersection of locus (1) and the line (4) will be also on the locus of the equation 'Z=.^ X^, J, 0=0; . . . (5) which, since the equation is in one variable, of finite degree, will represent a finite number of planes parallel to the ^z- plane, and therefore having a finite number of points of intersection with the line (4). Hence the locus of equation (1) is not a solid. Therefore, the locus of any algebraic equation in three vari- ables is a surface. 212. Curves. Traces of surfaces. Two surfaces intersect in a curve in space ; and since every algebraic equation in solid analytic geometry represents a surface, a curve may be represented analytically by the two equations, regarded as simultaneous, of surfaces which pass through it. Thus it has been seen that the equations y = b^ z = c separately rep- resent planes, but considered as simultaneous represent the straight line which is the intersection of those planes. But by the reasoning of Art. 41, the given equations of a curve may be replaced by simpler ones which represent other sur- faces passing through the same curve. In dealing with curves it is often useful to obtain, from the equations given, equations of cylinders through the same curve ; z.e., it is generally useful to represent a curve by two equations each in two variables only. Example : The curve of intersection of the two surfaces, (1) a;2 + ?/2 ^ ^2 _ 25 = and (2) x"^ + 3/2 - 16 = 0, 348 ANALYTIC GEOMETRY [Ch. II. is also the intersection of the surfaces a;2 + ^2 + ;22 _ 25 - (x2 + 2/2 - 16) = 0, i.e., z=±^, (3) with the surface (2). The curve is therefore composed of two circles of radius 4, parallel to the a:y-plane at distances + 3 and — 3 from it. Conversely, the curves of intersection of a surface with the coordinate planes may be used to help determine the nature of a surface. These curves are called the traces of the surface. Thus, the surface x^ + ^^ + ^^ _ 25 has the traces on the ^^-plane, where a; = 0, ?/2 -|- ^^ = 25 ; On the 2a;-plane, where y = 0, a:^ + ^2 _ 25 ; on the a:^-plane, wdiere ^ = 0, x^-\- y'^= 25. Each of these traces is a circle of radius 5, about the origin as center ; the surface is a sphere of radius 5 with center at the origin. Since three surfaces in general have only one or more separate points in common, the locus of three equations, con- sidered as simultaneous, is one or more distinct points. 213. Surfaces of revolution. Analogous to the cylinders are the surfaces traced by revolving any plane curve about a straight line in the plane as axis. From the method of formation, it follows that each plane section perpendicular to the axis is a circle, — the path traced by a point of the generating curve as it revolves ; and the radius of the circle is the distance of the point from the axis in the plane before revolution begins. These facts lead readily to the equation of any surface of revolution, as a few examples will show. (a) The cone formed hy revolving about the z-axis the line 2a;+ 3^ = 15. . . . (1) 212-213.] SURFACES 349 Any point P of the line (1) traces during the revolution a circle of radius ZP, parallel to the a:?/-plane. The equa- tion of that path is x^ + y'-^ ZPI z Fig. 151. But in the a;s-plane, before reA^olution is begun, LP is the abscissa x oi P ; hence, by equation (1), 15-32! LP = x = 2 so that the equation of the path of P is a? + f=(l^jzlll\ (2) But P is any point of line (1); hence equation (2) is sat- isfied by every point of the line, and represents the surface generated by the line, which is the required conical surface. (^) The sphere formed by revolving about the y-axis the circle 2:2-1-^2^25. . . . (3) In this case, any point P of the curve traces during the revo- lution a circle of radius iVP, parallel to the 2a:-plane. The equation of this path is therefore a;^ -j- ^2 ^ JSfP^^ 350 ANALYTIC GEOMETBT [Ch. n. Fig. 152. But in the ^ry-plane, by equation (3) NP = x= V25^2. Hence, substituting above, 2^2 _|_ ^2 _ 25 _ ^2^ i.e., x^+if + z^=2b', (4) wliich is the equation of the required spherical surface. ((?) The surface formed hy revolving about the x-axis the curve 2=(a;-l)(a:-2)(2^-3) [cf. Art. 37]. ... (5) Any point P of the generating curve traces a circle parallel to the ?/2 -plane, with a radius MP equal to the 2-abscissa in equa- tion (5). Hence the equation of its path is o B cW \M -X y2 -I- ^2 ^ ]^^ i.e., if -\- z^ = (x ~ 1)2 (2:-2)2(a^-3)2; . . .(6) which is the equation of the required surface. (67) Of the various surfaces of revolution those of particular interest are generated by revolving about their axes the various conic sections, giving the cones, spheres, paraboloids, ellipsoids, and hyperboloids of revolution. / y Fig. 153. 213.] SUBFACES 351 The student may verify the equations of the following surfaces ; * The sphere : with center at the point (a, 5, (?), and radius r, (^^-ay-\-(i/-by-i-(z-cY = r^; ... (7) with center at the origin, and radius r, ^2 + ?/2 4- 2;2 = r\ ... (8) The cone: the surface generated by the right line z = mx-\-c, rotated about the ^-axis, x^-\-f = ^^ -/)' . .... (9) The oblate spheroid : the surface generated by the ellipse — + — = 1, rotated about the minor axis, a^ b^ /Y>u 01^ ^^ -2 + ^-2 + h=^- ■ • • (10) a^ a'^ 0'^ The prolate spheroid : the surface generated by the ellipse — + — = 1, rotated about the major axis, /v>4 /}/^ n2t ' 1 + 1 + ^=1- • • • (") The hyperboloid of one nappe : the surface generated by the hyperbola — — ^ = 1, rotated about the conjugate axis, a^ 0^ /yii nia /ya ^ + ^-1=1- • • • (12) a^ a^ 0^ The hyperboloid of two nappes : the surface generated by the M^perbola 7-, — '-^ = 1? rotated about the transverse axis, 0^ a^ /yta 01^ ^y^ * See Chap. IV, where diagrams are given for the corresponding cases of the general quadric, with elliptical instead of circular sections. 352 ANALYTIC GEOMETRY [Ch. II. 213. The paraboloid of revolution : the surface generated by the parabola x^ = -^ pz^ rotated about its axis, x^ -\- y^ = ^ pz. . . . (14) EXAMPLES ON CHAPTER II What is the locus of each of the following equations? 1. ^2 - 6 a: + 9 = 0. 4. aj;2 + hxy + cy^ = 0. 2. 2a: + 4 = 0. 5. 4:yz + 6 ij - Sz + 1 = 0. 3. x^-2xy + y'^-\-2x-2y + l=0. 6. z'^-9y = 9. What are the curves of intersection of the surfaces represented by the equations 7. ?/ + 3 = 0, 3x2 + 3^2 + 3^22 ^20? 8. x^ -y^ = 0, z = a'> 9. x2 + ?/2 + 2-2 ^ 9, 4:r2 + ?/2 = 4? 10. 9(z2 + y2)_ .2^o5_10~, 5 =±5? 11. 3:r2-4y2_~2^i2, ^ + |^!=1? Determine the projections upon the coordinate planes of the following surfaces : 12. a;2 + ?/2 + 4 s2 = 25 ; 13. Sx^ - 4:y^ - z^ = 12. Find the equation of 14. the paraboloid of revolution one of whose traces is ?/2 = — 5 x + 3. 15. the curve of revolution one of whose traces is y = — 5 x + 3 and whose axis is the axis of y. Find its vertex. 16. the oblate spheroid one of whose traces is — + — = 1. <- o V z 17. the prolate spheroid one of whose traces is ^ + — - = 1. 18. the surface of revohition whose axis is the axis of x and one of whose traces is x'^y — 1 = 0. 19. the hyperboloid of two nappes one of whose traces is 16 x^ -9^2:=!. 20. the sphere described about the major axis of the ellipse 4 a:2 + 9 3/2 — 24 a; = as diameter. CHAPTER III EQUATIONS OF THE FIRST DEGIJEE Aoc + Bij + Cz + D = () PLANES AND STRAIGHT LINES I. The Plane 214. Every equation of the first degree represents a plane. A plane is a surface such that it contains every point on a straight line joining any two of its points. Let P^ = (a;^, ?/-^, 2j) and P^ = (x^, y^^ z^ be any two points of the surface whose equation is ^ic + J5t/ + Cs + 1> = 0, . . . [16] so that Ax^ -\- By^ + Cfe^ + i> = . . . (1) and Ax^ + By^ + 0b^ + D = 0. . . . (2) Now, if Pg = (2^3, ^3, ^3) be any point on the straight line from Pj to P^ at a distance d^ from P^ and d^ from P^^ then, by Art. 205, ^ "~ C^i + 6?2 ^ C?l + t?2 ' ^ ~ t^l + d^ ^' ^ But this point lies on the surface represented by equation [16]; for, substituting its coordinates from (3) in equation [16], the latter becomes Cti -p Clt) Cl-i -f" (to TAX. AN. GEOM. 23 353 354 ANALYTIC GEOMETRY [Ch. III. which is a true equation, since each parenthesis vanishes separately by equations (1) and (2). Hence every point of the line P^P'^ is on the locus of equation [16], and that locus is therefore a plane. Every algebraic equation of the first degree in three variables represents a plane. 215. Equation of a plane through three given points. The general equation of the first degree, Ax-vBy-{-Cz + D = Q, . . . (1) has only three arbitrary constants, viz. the ratios of the coefficients. If three given points in the plane are then these ratios may be found from the three equations, Ax^ + By^ + Cfej + i> = 0, ■ Ax^ + %2 + ^^2 + -^ = 0, • . . . (2) Ax^ + %3 + C^3 + 2) = 0, . considered as simultaneous. In solving equation (2) for the required ratios, two special cases may occur : {a) The value of one of the coefficients may be zero, then the ratios determined must not have that coefficient in the denominator. E.g.., if J> = 0, solution A Ti n A Ti should not be made for — , —, — , but for — , — • (say). jj U JD (5) The equations may differ only by constant factors, then the three equations have an infinite number of solutions. This is explained by the fact that the points are on a straight line, and any plane through the line will pass also through the points. 216. The intercept equation of a plane. A plane will in general cut each coordinate axis at some definite distance 214-217.] PLANES AND STRAIGHT LINES 355 from the origin, and this distance is called the intercept of the plane on the axis. If a, 6, c be the intercepts on the x-, ?/-, and 2-axes, respectively, of the plane whose equation is Ax + By + (7^ + i> = 0, . . . (1) then the points (a, 0, 0), (0, ^,0), (0, 0, c) are points of the plane, and therefore (cf . Art. 215) J.a + i> = 0, Bh + D = 0, Cc^D = 0, B • • • G Hence equation (1) may be written i.e. a (2) I.e., a h c - + ^ + --1- [17] and this is the equation of the plane in terms of its intercepts. 217. The normal equation of a plane. A plane is wholly determined in position if the length and direction be known of a perpendicular to it from the origin ; and this method of fixing a plane leads to one of the most useful forms of its equa- tion. Let OQ be the perpendicular from the origin to the plane ABO^ let p be its length, always considered as positive, and let «, /3, 7 y be its direction angles. Let P = (x, ?/, z) be any point of the plane, and draw its coordinates OM, MM', M'P. Then, projecting upon OQ^ Fig. 154 356 ANALYTIC GEOMETRY [Ch. III. proj. OMM'P = proj. OP, hence proj. 0M+ proj. MM' -\- proj. M' P = proj. OP, that is, a?cosa + t/cosp + 2: cos-y =i>. . . . [18] This is called the normal equation of the plane. There are two special cases to be considered : (1) If the plane is perpendicular to a coordinate plane, e.g., to the :?^y-plane (cf. Art. 210), then 7 = 90°, cos 7=0, and equation [18] reduces to a; cos « + ?/ cos /3 = j9. . . . [19] (2) If the given plane is parallel to one of the coordinate planes, e.g., to the a:y-plane (cf. Art. 209); then «=/3=90°, 7 = 0°, and eq. [17] reduces to z=p. . . . [20] 218. Reduction of the general equation of first degree to a standard form.* Determination of the constants a, b, c, i>, a, p, 7. I. Intercept form. In Art. 216 a method has been indicated for reducing the general equation Ax + By^Cz + D=0 . . . (1) to the intercept form. Since the points (a, 0, 0), (0, h, 0), and (0, 0, c) are on the plane (1), it follows that the inter- cepts are D.I) D .0. " = -T ^ = -B^ ' = -c' • • • (^) II. Normal form. If equation (1) and the equation X cos a -\- y cos fi -{• z cos 7 — _p = . . . (3) represent the same plane, then their first members can differ * The reduction of this article gives a second proof that the general alge- braic equation of first degree always has for its locus a plane. 217-219.] PLANES AND STRAIGHT LINES 357 only by a constant factor, m (cf . Art. 203, eqs. [5] ; also Art. 58); therefore mA = cos a, 7nB = cos yS, m = cos 7, mD = — p, but, by [4], cos^ « + cos^ jS + cos^ 7 = 1, hence m^(^A'^ -{- B^ + 0^} = 1, and m = VWTW+c^ Then cosa^ - ^ cosp = ^ V^2 + ^2 + c^2 V^2 + ^2 + (72 r' — /> cos 7 = > p V^2 + ^2 + ^.2 V^2 _j. j52 4. (72 Equation (1) written in the normal form is then A , B [21] g= , _ ; ... (5) VA2 + ^2 + C'2 VJ.2 + ^ + 6'2 therefore, to reduce equation (1) to the normal form, it is nec- essary only to transpoBe the constant term to the second mem- ber of the equation^ and then divide both members by the square root of the sum of the squares of the coefficients of the variable terms. The sign of the radical is determined by the fact (Art. 217) that p is taken positive ; hence, the sign of the radical is the opposite of the sign of the constant term. 219. Tlie angle between two planes. Parallel and perpen- dicular planes. The angles formed by two intersecting planes are the same as the angles formed by two straight lines perpendicular to them respectively; i.e.^ is the same 358 ANALYTIC GEOMJ^TRT [Ch. IIL as the angles between the respective normals from the origin to the planes. If A^x + B^y + O^z + I)^ = 0, . . . (1) and A^x + B^y + C^z + i>2 = 0, . . . (2) be two planes, then the direction cosines of their normals are respectively (eqs. [21]) COS«i = — ^^ , C0S)8i= — ^ , C0S7l=- ^ V^i2+^i^+CV y/A{^ 4- Bi^ + Ci'^ V^i^ + Bi^ + C{^ ' cosctg^ '^ , etc., y/A^^+Bi^+Ci^ and by equation [4], if be the angle betAveen the two planes, and hence between the two normals, cos 9 33 ^l^2 + Bl g 2jL Cie2 . . . p22-| There are two cases of special interest. I. Parallel planes. If the planes (1) and (2) are parallel, their normals from the origin will have the same direction co- sines, and differ only in length ; therefore, by equations [20], the equations of the planes must be such that the coefficients of the variable terms are the same in the two equations, or can be made the same by multiplying one equation by a constant. In other words, if the planes (1) and (2) are parallel, then X = f = ^' • • • [23] -^2 ^2 ^2 and the plane Ax -\- By -{- Cz -\- K= . . . (3) is parallel to the plane Ax + By + Cz + B = 0, . . (4) for all values of the parameter K. 219-221.] PLAJ^ES AND STRAIGHT LINES 359 II. Perpendicular planes. If the planes (1) and (2) are perpendicular to each other, then cos ^ = 0, and AiA2 + BiB.^+ CiC2 = 0; . . . {24:'] and conversely. 220. Distance of a point from a plane. Let A = C'^i'^i'^i) be a given point, and Ax-{-B^-{-Cz-\-D=^0 . . . (1) a given plane. The perpendicular distance of P^ from the plane is equal to the distance from the plane (1) to a parallel plane through the point; i.e., is equal to the difference in the lengths of the normals, from the origin, to these two parallel planes. The parallel plane through Pj has for its equation by Art. 219, equation (3), Ax + Bt/ -\-0z = Ax^ + By^ + Cz^ . . . (2) By [21], the lengths of the normals to planes (1) and (2) are, respectively, ^^ -I> , _ Ax, + By, + Cz^ therefore ii d = p' — p hQ the required distance, a = ^^^i±^?yi±j^^i±R. . . . [25] V^2 + B^ + C^ In formula [25], the sign of the radical is taken opposite to the sign of D (Art. 218) ; and the sign of d shows on which side of the given plane lies the given point. II. The Straight Line 221. Two equations of the first degree represent a straight line. Every equation of first degree represents a plane 360 ANALYTIC GEOMETRY [Ch. III. ,(Art. 214), and two equations considered as simultaneous represent the intersections of their two loci (Art. 39). Therefore since tw^o planes intersect in a straight line, the locus of the two simultaneous equations of first degree, A^x + B^y + C^z + D^ = Q, A^x + B^y + C^z + D^ = 0, . . . (1) is a straight line. As suggested in Art. 212, it is generally more simple to represent the straight line by equations in two variables only, standard forms^ to which equation (1) can always be reduced. 222. Standard forms for the equations of a straight line. (a) The straight line through a given point in a given direction. Let P^ = (x^^ ?/j, z^ be a given point, and a, y8, 7 the direc- tion angles of a straight line through it. Let P = (x^ y, 2) be any point on the line, at a distance d from Py Then by equation [6], d cos a = X — x-^^ d cos P = y — yi-, d cos ry — z — z-^^ . . . (1) hence x^^oo,^y-j^^z-^ _ _ _ ^^^-^ COS a COSp COS7 which are the equations of a straight line in the first standard form, called the symmetrical equations. (6) The straight line through two given points. Let P^ = (xy y^ z^ and P^ = {x^., y^, z^ be the given points. Any straight line passing through P^ has [26] for its equations. If the line passes also through P^^ then ^2 ~ ^1 _ Vi ~ Vx _ ^2 ~ ^1 . fs\\ cos a ~ cos y8 ~~ cos 7 ' * * * ^ ^ and hence from equations [26] and (2), by division to eliminate the unknown direction cosines. a?2 - ^1 2/2 - 2/1 ^2 - ^1 [27] 221-222.] PLANES AND STRAIGHT LINES 361 These are the second standard forms for the equation of a straight line. (c) The straight line ivith given traces on the coordinate planes. One of the simplest set of planes for determining a straight line is a pair of planes through the line and perpen- dicular respectively to the coordinate planes (cf. Art. 212). Then the equation of these planes will be the same as the equations of the traces of the line on the corresponding coor- dinate planes (Art. 210). Thus, if the equation of the traces of a given line upon the zx- and ^2-planes are, respectively, y = nz -\- d^ then, considered as simultaneous, these are also the equa- tions of the given line in space. In Fig. 155 the given traces are ABL' in the 0a;-plane, and CDN' in the 2/2-plane ; P is any point in the given straight line, and §, i2, S are the points where the line pierces the xy-^ yz-., 2!a;-planes, respec- tively. Then it is clear that in equations [28] m = tanZ O^^ji. h = OB, n = tanZ OOB, d= OB. Fig. 155 (3) Also, since, by equations [28], bn m m 00=-i, OS = ^" - '^"' , n n 362 ANALYTIC GEOMETRY [Ch. III. therefore the points where the given line pierces the coordi- nate planes are Q^(_b,d,0), B^fo, *?i=*!!,_A\ fef*Ji:z*^, 0, -^. (4) ^ \ m mj \ n nj 223. Reduction of the general equations of a straight line to a standard form. Determination of the direction angles and traces. I. Third standard form: traces. The traces of a straight line have the same equations as have the planes of projec- tion of the straight line upon the coordinate planes, respec- tively. They may be obtained, therefore (Art. 210), by eliminating in turn each of the variables ^, ^, x from the given equations. This may be illustrated by a numerical example. Given the equations 3a; + 2^ + 2 -5 = 0, a:-f 2^ -2^ = 3, . . . (1) representing a straight line. Eliminating z, y, and x, successively, the equations 7:r + 4?/ -13 = 0, 2a; + 32 -2=0, 4^-7^-4 = ... (2) are obtained, each representing a plane through the given line and per- pendicular to a coordinate plane. Therefore these equations are also the equations of the traces of the line, in the xy-, zx-, and i/s-planes, respectively. II. First standard form : direction angles. The method of reducing the general equations of a straight line to the first standard form, and finding its direction angles, can also be illustrated by a numerical case. Considering still the line whose equations are (1) above, and w^hose traces are given by equations (2) ; and taking the equations of any two of its traces, e.g.., 2a: + 3s-2 = 0, 42/-7s-4 = 0; . . . (3) 222-224.] PLANES AND 8TEAIGHT LINES 363 these have one variable, z, in common. Equating the values of this common variable from the two equations, gives _ -2.r + 2 _ 4y-4 3 7 which may be written, to correspond with equations [26], g -Q ^ a;- 1 ^ y - 1 ^ ^ /^A 1 3 7 * • • • V y i — 2 ? Now, although the denominators 1, — f, | of equation [4] are not du'ection cosines of any line, yet, by equations [5], they differ from such direction cosines only by the factor Vl + f + ft = iVlOl. Rewriting equations (4) in the form X — 1 y — ^ z — 6 (5) loi VlOl VlOl it corresponds entirely to equations [26]. Therefore the line passes through the point (1, 1, 0), and its direction angles are given by the relations 6 o 7 ^ cos a = , cos p = — , cos y VlOl VlOl VlOl The method given above is evidently perfectly general. 224. The angle between two lines ; between a plane and a line. If the equations of two straight lines be written in the form ^ - ^1 ^ ,y - ^1 ^ ^ - ^1 ^ . . . (1) ^ — ^1 ^ y — Vi ^^ — H ... (2) «2 ^2 ^2 then by Art. 223, II, their direction cosines are, respectivel}^ cos «j = — t cos ^2 = cos /3i = ^ etc., ... (2) 364 ANALYTIC GEOMETRY [Ch. 111. and therefore, by equation [10], the angle between the two lines is given by the equation coGe= ^1^.2 + &1&2 + C1C2 |-29] Again, the angle between the straight line ^'E^zJb. = y-::zli = ^^-zJi\^ . . c (3) a h c and the plane Ax^-By-\- Cz^I)=^ . . . (4) is the complement of the angle between the line (3) and the perpendicular to the plane (4) from the origin. Therefore, by equations [10] and [21], and Art. 223, II, the required angle is given by the equation sine^ aA+b B + cC ^ Conditions for perpendicularity and parallelism precisely like those of Art. 219 may be obtained from equations [29] and [30]. EXAMPLES ON CHAPTER III 1. Find the equation of a line through the points (1, 2, 3) and (3, 2, 1). 2. Find the equation of a plane through three points (1, 2, 3), (3, 2, 1), and (2, 3, 1). 3. Write the equations of the straight line through the point (1, 2, 3), and having its direction cosines proportional to V-S, 1, 2V3. 4. What are the traces of the line of Ex. 1 upon the coordinate planes? Where does the line pierce those planes? 5. Find the equations of a straight line through the point (1, 2, 3) and perpendicular to the plane x-\-2y + ^z = 6. Reduce to the intercept and normal forms, and determine which octant each plane cuts : 6. 2x-^y-z = 7; 7. 5y + 2z -1 = x. - 224.] PLANES AND STRAIGHT LINES 365 8. Reduce the equations of the line 2x -Zy -z = 7, 5 ^-f 2z -1 = X to the symmetrical form, and detei-mine its direction cosines. 9. Find the angle between the planes 2a:— 3?/ — 2 = 7, ^y-\-2z — l=x. 10. Find the angle between the line X + y + 2 z = 0, 2x-y-2z-l=0, and the plane 2tx-\-Qz — by + l = 0. 11. Write the equation of a plane parallel to the plane 2x-?/ + 7s-5 = 0, and passing through the point (0, 0, 0) ; through the point (-1, 1, -1). 12. Write the equation of a plane perpendicular to the plane and passing through the two points (3, 1, 2) and (0, ~2, -4). 13. Find the distances of the points (7, ~2, 3) and (3, 3, 1) from the plane 2a: + 5?/ — s — 9 = 0. Are they on the same side of the plane ? 14. At what angle does the plane ax + hy + cz -\- d = cut each coordi- nate plane? Each coordinate axis? 15. Find the equation of a plane through the point (1, 1, 1) and perpendicular to each of the planes 2a:-3?/ + 73 = l, x - y -2z = 2. 16. Write the equation of a plane whose distance from the point (0, 2, 1) is 3, and which is perpendicular to the radius vector of the point (2, -1, -1). 17. ^Vrite the equation of a straight line through the point (5, 2, 6) which is parallel to the line 2x-32 + ?/-2 = 0, a; + ?/ + 2 + l=0. 18. Find the traces on the coordinate planes of tlie line 2a;-32 + ?/-2 = 0, x + y + z + l=Q. 19. Prove that the planes 2 a; - 3 y + ^ + 1 = 0, 5 a; + 2 - 1 = 0, 19a;-3?/-43-5 = 0, have one line in common. 366 ANALYTIC GEOMETRY [Ch. III. 224. 20. What is the equation of the plane determined by the line and the point (5, 2, 6) ? 21. Show analytically that the locus of a point equidistant from three given points is a straight line perpendicular to the plane determined by those three points. 22. Derive equation [17] directly from a figure, without using equa- tion [16 j. CHAPTER IV EQUATIONS OF THE SECOND DEGREE QUADRIC SURFACES 225. The locus of an equation of second degree. The most general algebraic equation of second degree in three variables may be written Aoc^ + By^ + Cz^ + 2 Fyz + 2 Gxz + 2 Hocy + 2 Xa? + 2 3Iy + 2Nz + K = (i, . . . [31] Any surface which is the locus of an equation of second degree is called a quadric surface, and is of particular interest because of its close connection with and analogy to the conic sections. In fact, every plane section of a quadric is a conic, as may be easily shown as follows. By Art. 207, any plane may be chosen as a coordinate plane, and the transformation of coordinates to the new axes will leave the degree of equation [31] unchanged ; i.e.^ the new equation of the locus will still be of the form [31], though with different values for the coefficients. To find the nature of any plane section, choose the given plane as (say) the xy- plane of reference, and transform to the new axes ; the new equation will be of form (1). Then let z = 0. The equa- tion of the section of the quadric is Aa^ + By^-^ 'I Exy + 2 Lx + 2My + K=0; . . (1) and this, by Art. 175, represents a conic. 367 368 ANALYTIC GEOMETRY [Ch. IV. Moreover, the trace of the surface on any parallel plane, as 2 = «, is given by the equation Ax^ + Bi/ + ^ Hxy + 2(X + ah)x + 2QM+ aF)y + (CVH-2il[fa + ^)=0. ... (2) Now, by Arts. 177, 181, the loci of equations (1) and (2) are conies of the same species, and with semi-axes pro- portional; therefore their eccentricities are equal, and the curves are similar. Hence, all parallel plane sections of a quadric are similar conies. 226. Species of quadrics. Simplified equation of second degree. As will be seen in the following sections, quadric surfaces may be conveniently classed under four species. For, although different plane sections of any surface will in general be conies of different species, still the general form of the surface may be characterized most strikingl}^ by those plane sections which are ellipses, hyperbolas, parabolas, or straight lines. These species are called, respectively, ellip- soids^ hyperloloids, paraboloids^ and cones; and each species has special varieties, depending upon the nature of a second system of plane sections. To study these species it will be well to simplify the general equation of second degree as much as possible by a suitable transformation of coordinates.* A transformation of coordinates changing to a new rectangular system having the same origin as the old, by equations [14], will transform the given equation of second degree to A'x^ + B^y'^ + C'z^ + 2 F'yz + 2 a^xz + 2 Wxy + 2 L^x -^2M'y+2N'z + K=0, . . . (1) where A' ^ B\ "- N' are functions of the nine direction angles * Compare with Art. 176. 225-226.] QUADRIC SURFACES 369 ctj, «2i ••• of the new axes, which are limited by the six inde- pendent equations noted in Art. 207. These angles, therefore, may be so chosen that three additional conditions shall be fulfilled ; hence, so that the coefficients F', G', and R' shall vanish. Then the new equation of the quadric will be A'x^-{- B'i/-\- Q'z^+ 2 L'x -f- 2 M'y + 2 N'z +K= 0. (2) Now a second transformation may be made to a parallel system of axes through a new origin (A, k^ y), by equations [13], giving for the new equation A'x^ + B'y'^ + C'z^ + 2 L^'x + 2 M"y + 2N"z + K' ^0, (3) in which X'', M" ^ N", and K' are functions of the coordi- nates A, ^, and j ; and these coordinates may be chosen so that X", M'\ and N" will vanish, giving for the simplified form of the equation of the given quadric, A'x^ + B^y'^ + C^z^ + K' = ^. . . . (4) It may happen, however, that the choice given above for the direction angles ce^, a^^ •••, of the new axes is such that the coefficient of one more term of second degree, as (7', will also vanish ; then equation (4) would reduce to A'x^ + B'y'^ + K' = 0, ... (5) and the surface is a cylinder (Art. 210). Again, if also X", M" ^ N" are not independent, and the values of A, ^, j as given above are therefore indeterminate, then 7i, ^, j may be chosen so that, for example, L", M"^ and K' shall vanish ; and the equation of the quadric becomes Ax^^-B'y'^-^'lN"Z=0^ ... (6) * If the coefficients of two quadratic terms vanish, as B' and 6*', a change of origin first, then of direction of axes, may be chosen so that the equation will reduce to the form (6). TAN. AN. GEOM. 24 ^2 = 52= _:^, ^= _^. . . . (1) 370 ANALYTIC GEOMETRY [Ch. IV. The two forms of tlie quadric, not already discussed,* have therefore for their equations, when simplified (dropping the accents), Ax' + Bf + Oz^ + K=0, . . . [32] and Ax^-^Bf + 2M = 0. . . . [33] A center of a surface is a point such that it bisects every chord of the surface which passes through it. It is clear that the locus of equation [32] is a central quadric, while the locus of equation [33] is non-central (cf. Art. 178). 227. Standard forms of the equation of a quadric. For convenience of discussion, the intercepts of the locus of equation [32] on the coordinate axes may be represented by a, b, c, respectively, so that T "- B' '^- o Then, since A, B, C, and K cannot be all of the same sign, there will be three types of equation [32], according to the signs of A, B, (7, and K; viz.: ^2+52-^2-^' • • • y^^j a^ b^ c^~ * • • ^^ Similarly, equation [33] may be written for convenience in the typical forms ^2 ^^ j2 — ^' • • • y'^J a2 62-^' • • • y'') * An exceptional case occurs where the general equation can be factored into linear factors, and therefore represents two planes. 226-228.] QUADBIC SURFACES 371 wherein, however, a and b are no longer intercepts as in (2), (3), and (4). Again, if the equation [32] has its constant term zero, it may be written in two typical forms, a^ h^ c^ (8) These seven equations are standard forms of the equation of second degree, and will be discussed in turn. o o 2 228. The ellipsoid: equation ^ + f-2 + ^ = l- ^'^^^^ ^^® equation ... [34] ^^^4.^ = 1 a2 "^ &2 -^ c2 the following properties of its locus may be derived : (1) The traces on each coordinate plane are ellipses, having Fig. 156 372 ANALYTIC GEOMETRY [Ch. IV. the semi-axes a and h in the ir?/-plane, h and c in the yz- plane, and c and a in the ;sa;-plane. (2) The traces on planes parallel to any coordinate plane are similar ellipses (Art. 225). (3) The equation may be written y ^2 ~^ = 1 hence for a plane section parallel to the «/;3-plane, the semi- axes are real if the value of x lies between — a and -{- a, imaginary if beyond those limits, and zero ii x= ± a. More- over, the length of the axes diminish continuously from the values h and 45°, the conic is an hyperbola (cf. Part I, Art. 48). NOTE C The special cases of the conies. The locus of the second degree curve has been seen to have three species, according as el. If e = 0, then, since h is defined by the equation 6"2 = a^(l — e^), b = a, and the curve is an ellipse with equal axes, i.e., it is a circle ; in this case, also, the directrix is at infinity and the focus at the center, for the equa- tion of the directrix is x = -, and the distance from the center to the e focus is ae (cf. Part I, Arts. 110, 116). ® 384 APPENDIX Again, suppose the focus F to be on the directrix. Then, if P is any point of the locus, and LP perpendicular to FD, FP = e'LP, . . . (1) and smZPFL=^ = l; , . . (2) l' P e hence the angle PFL is constant, with two supplementary values for a given value of e. The locus consists therefore of two straight lines intersecting at P, and equation (2) shows that : if e > 1, the lines are real and different ; if e = 1, the lines are real and coincident ; and if e < 1, the lines are imaginary, and the real part of the locus consists of the point F. Suppose now the directrix, with the focus upon it, to be at infinity ; then, if e > 1, the locus is a pair of parallel lines. These results agree with those akeady summarized in Art. 182. NOTE D Sections of a cone made by a plane. The following proposition is due to Hamilton, Quetelet, and others (see Taylor's Ancient and Mod- ern Geometry of Conies, p. 204). If a right circular cone is cut by a plane, and two spheres are inscribed in the cone and tangent to this plane, then the section of the cone made by the plane is a second degree curve (cf. Part T, Arts. 48, 175), of which the foci are the points of contact of the spheres and the plane, and the directrices are the lines in which this plane intersects the planes of the circles of contact of the spheres and the, cone. Construction : Let 0-VW be a right circular cone cut by the plane HK in the section RPSQ, P being any point of the section. Inscribe two spheres, C-ABF and C'-A'B'P, whose circles of contact with the cone are AEB and A'E'B', respectively, and which are tangent to the plane HK in the points F and F. Through P draw the element OP of the cone, cutting the circles of contact in the points E and E'. Also pass a plane MN through the circle AEB, and therefore perpendicular to the axis OCC oi the cone ; it will intersect the plane HK in a straight APPENDIX 385 line GDL, which is perpendicular to the straight line F'F. Draw PL perpendicular to GDL. P Then PL makes a constant angle (=Z F'DA) with the plane MN [since PL is parallel to F'F^, and, if p represents the distance from the point P to the plane MN, p=PL sin e. . . . (1) Also PE, being an element of the cone, makes a constant angle a with the plane MN, and p =PE sin a. . . . (2) Again, since tangents from an external point to a sphere are equal, PE=PF. . . . (3) TAN. AN. GEOM. 25 386 APPENDIX Hence, from equations (1), (2), and (3) PF sin " PL sin a = e, a, constant, . , . (4) i.e., the ratio PF : PL, for every point P of the section SPRQ, is constant, and (Part T, Arts. 48, 175) the section is a second degree curve, with a focus at F, directrix GDL, and eccentricity — . sm a Similarly, F' is the other focus, and the line of intersection of the planes HK and A'E'B' is the other directrix of the conic SPRQ; hence the theorem is established. Moreover, the plane VW, being perpendicular to the axis of the cone, and OVW, being a section made by a plane passing through the axis, a = Z OVW, and is constant for a given cone, while = Z OSR, and varies only with the plane HK. Hence the eccentricity varies with the inclination of the plane HK, and there are the three following cases : if ^ < a, then e < 1, and the section is an ellipse ; if ^ = «, then e = 1 , and the section is a parabola ; if ^ > a, then e > 1, and the section is an hyperbola. Again, if the cutting plane HK passes through the vertex of the cone, then the focus F is on the directrix GDL, and the section will be either a pair of straight lines or a point : if ^ < a, the section is a point, the vertex of the cone. M = CL, the section is a pair of coincident straight lines, an element of the cone ; ii6> a, the section is a pair of intersecting straight lines, two elements through the vertex (cf. Note C). It is, of course, evident that for every elliptic section of the focal spheres both lie in the same nappe of the cone, and touch the plane of the section (HK) on opposite sides; while for every hyperbolic section these focal spheres lie one in each nappe of the cone, and both on the same side of the plane of the section. In the above proof, for the sake of simplicity, a right circular cone was employed; it is easy to show (see Salmon's Conic Sections, p. 329) that every section of a second degree cone (right or oblique) by a plane is a second degree curve. APPENDIX 387 The demonstration just given shows also that the parabola is a limit- ing case of an ellipse (cf. Note E). ^_ NOTE E Parabola the limit of an ellipse,* or of an hyperbola. If a vertex and the corresponding focus of an ellipse remain fixed in position while the center moves further and further away, the major axis becoming infinitely long, then the form of the ellipse approaches more and more nearly to that of a parabola having the same vertex and focus. This is easily shown as follows : The equation of the ellipse referred to its major axis and the tangent at its left-hand vertex, as coordinate axes, is (Part I, Art. 112) {x - ay if _ «2 "^ 62 ~ ■"' which may be written in the form 2 2&2 y^ = X 62 a a^ If now the fixed distance OF be represented by p, then p = OF = OC - FC = a - \/«2 _ b% whence h'^ = 2 ap — p'^ ; 62 2 p /)2 (i) (2) therefore 262 2»2 =r 4 » — , and a a a^ a a' * This fact is of importance in astronomy in connection with the behavior of comets. 388 APPENDIX Substituting these values in equation (2) it becomes .^-(*^-'-i^>-(¥-a^' • • • (^) and the limit of this equation as a approaches oo, p remaining constant, is ?/2 = 4^a:; . . . (4) which is the equation of a parabola, and the proposition is proved. In the same way it may be shoAvn that the parabola is the limit to which an hyperbola approaches when its center moves away to infinity, a vertex and the corresponding focus remaining fixed in position (cf. also Note D). NOTE F Confocal conies. — Two conies having the same foci, F^ and Fg, are called confocal conies. Since the transverse axis of a conic passes through the foci and its conjugate axis is perpendicular to, and bisects, the line joining the foci, therefore confocal conies are also coaxial,* i.e., they have their axes in the same lines. If the equation of any one of such a system of conies is and if X is an arbitrary parameter, then the equation + 7^=1 ... (2) a'^ + X h^ + X will represent any conic of the system. For, a and b being constant, and a'^b, equation (2) represents ellipses for all values of X between co and — b% hyperbolas for all values of X between — //^ and — a% and imaginary loci when X< — a^; moreover, the distance from the center to either focus for each of these curves is V(a2 + A) - (62 + X), which equals Va^ — b^, and is therefore constant. The individual curves of the system represented by equation (3) are obtained by giving particular values to X, each value of X determining one and but one conic. If any one of these conies is chosen as the * Coaxial conies are, however, not necessarily confocal. / APPENDIX 389 fundamental conic, and represented by equation (1), then each of the other conies of the system may be designated by its appropriate vahie of X. .^ ^^ ---^ / ^^N^/\ \ ^ "" ^*«*^/ /\y^ /xV'^ y\^ ~~~^— =^Cl/H^ ' y\ / /Or* "^ '"" TttC 'x \ ( (\ \ "' 1 ii \ \ Y^^--- Iw 1^ "_Jj^ V Through any assigned point, P^ = (a:j, y^^ of the plane, there passes one ellipse and one hyperbola of the system represented by equation (2). For substituting the coordinates x^ and y^ of P^ in equation (2), it gives the quadratic equation + Vx + A &2 + X = 1, (3) for the determination of X. Equation (3) gives two values of A, hence two conies of this confocal system pass through P^. That one of these is an ellipse and the other an hyperbola is shown as follows : the quad- ratic function in A + yi' a2 + A 62 + A is negative when A = + co, and, as A decreases from + go to -co, this function becomes positive just before \--l\ negative again just after \ = -W-^ and positive again just before A = -a'^; hence, of the two roots of equation (3), one lies between - }P- and co, and the other between — a2 and — 6^; and therefore of the two confocal conies which pass through Pj, one is an ellipse and the other an hyperbola. ^loreover, the two confocal conies which pass through any given point, as P^ = {x^, ?/,), of the plane intersect at right angles. This is easily seen geometrically thus: connect P^ with the foci F^ and P^, then the tangent P^T^ to the 390 AFFENBIX hyperbola through Py bisects the interior angle between F^P^ and F^P^, while the tangent P-^Tc, to the ellipse through this same point bisects the external angle formed by these two lines (cf. Part I, Arts. 148, 163) ; these tangents are therefore at right angles, hence (cf. Part T, Art. 100) the conies intersect at right angles. This fact could also have been readily proved analytically by compar- ing the equations of the two tangents. Remark 1. It is easily seen that as X approaches — h^ from the positive side, the ellipses represented by equation (2) grow more and more flat (because the length of the semi-minor axis VA- + A approaches 0), approaching, as a limit, the segment F^F^ of the indefinite straight line through the foci. On the other hand, if X approaches — h^ from below, then the hyperbolas grow more and more flat, approaching, as a limit, the other two parts of this line. Again, if A, approaches — a^ from above, the hyperbolas approach the z^-axis as a limit. Remark 2. Since through every point of a plane there passes one ellipse and one hyperbola of the confocal system represented by equation (2), and but one of each, therefore the two values of X which determine these two curves may be regarded as the coordinates of this point; they are known as the elliptic coordinates of the point. If the rectangular coordinates of a point are known, the elliptic coordinates are easily found by means of equation (2). E.g., let Pi= {x^, yi) be the point in question, then the elliptic coor- dinates of Pj are the two values of X, which are the roots of equation (3). So, too, if the elliptic coordinates are given, the Cartesian coordinates can be found. Remark 3. The above observations concerning confocal conies are easily extended to confocal quadrics, i.e., to quadric surfaces whose principal sections are confocal conies. They are represented by the equation + j!- + -^i- = i. + A b^-\-X c2 + A Date Due ' -■ , ■ ~*- -^» H6B2o'^H, ^ SB27'&aj^ f) PRINTED IN U. S. A. 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