“tt THE UNIVERSITY OF ILLINOIS LIBRARY I+655 @ib Gee Glie. ue af ATC BS ay b ugh WAT HERATICS Return this book on or before the Latest Date stamped below. University of Illinois Library DEC 29 1961 Lt41—H41 - ce ee “id Baas i ant Eee ere + igen ode aed emi : y § aa afc; r A TREATISE ‘ ON f ae i ae PLANE CO-ORDINATE GEOMETRY; e a é ame, $ | Ty “i ’ OR, THE pet pats - APPLICATION OF THE METHOD OF CO-ORDINATES be a “ | THE SOLUTION i Ahi ‘ - - PROBLEMS IN PLANE GEOMETRY. ce oa \? 2 ae re) , Rn ek PARC I. a BY THE REV. M. O’BRIEN, | OPESSOR OF NATURAL FHILOSOPHY AND ASTRONOMY IN KING’S COLLEGE, LONDON, AND LATE FELLOW OF CAIUS COLLEGE. eDEILGH TONS, CAMBRIDGE; WHITTAKER & CO., LONDON. 1844. CAMBRIDGE: PRINTED BY METCALFE AND PALMER, TRINITY-STREET. Pie ae Tae ow ae Tee ee Saheb. RK: : ike MAR 25 ‘28 PREFACE. THE subject treated of in the following pages is usually styled Analytical Geometry, but its real nature seems to be better expressed by the title Co-ordinate Geometry, since it consists entirely in the application of the Method of Co- ordinates to the Solution of Geometrical Problems. The present Treatise, in which we shall confine our at- tention to figures and curves in one plane, will consist of two parts: the first part is all that is at present published; it contains the application of the method of Co-ordinates to Right Lines, Circles, and Conic Sections. In the second part, the properties of Curves in general, with reference to Tangents, Asymptotes, Singular Points, Curvature, &c., will be investigated, without assuming a knowledge of the Diffe- rential Calculus on the part of the Student. A Historical Account of the subject, and a large collection of Problems will be added. The complete analysis given in the Table of Contents renders it unnecessary to say much here respecting the plan pursued in the ‘Treatise. In the first chapter the meaning of the signs = + and -, and the nature of negative and imaginary quantities, are fully explained, on principles which seem to combine generality and simplicity. The difficulties fo ye? grey SU0CVUD 1V PREFACE. which are supposed to beset the foundations of Algebra, are partly due to the indistinctness of the definitions usually given of algebraical symbols. In the subsequent chapters, the Author has endeavoured to adhere to a uniformity of method which, he hopes, will render it easy to acquire and retain a knowledge of the subject. He has also made use of symmetrical equations as much as possible, as there can be no doubt that many advantages are lost, and none gained, by want of attention to symmetry in analytical processes. The properties of Conjugate diameters are investigated by means of the angle called the Eccentric Anomaly in Astro- nomy, but the same properties are deduced without making use of this angle in Chapter XI. Several geometrical illustra- tions are given in notes, and various Problems and Examples, many of which are taken from the Senate-House Problems and other sources are added at the end of each chapter. CAMBRIDGE, March, 1844. TABLE OF CONTENTS. CHAPTER I. Meanine of the Signs=+-. Nature of Negative Quantities. Multiplica- tion by a Negative Quantity. Impossible Quantities. Meaning of the Sign —%. Sense in which the Signs = + — are used in the present subject. ARTICLE 1. Meaning of the Sign =. 2,3. Meaning of + and -. + Examples. toe A YS VY YX, 6. Our definition of X — Y does not require X to be greater than Y. 7 Definition of a Negative. 8. A positive quantity what. 9, 10. Of the Notation —Y, and + Y. Mizg id. —(-Y)=Y, -(X-Y)=Y- xX. 14. To add —Y is the same thing as to subtract Y, and to subtract -Y the same thing as to add Y. 15, 16. General method of extending the meaning of a Notation to cases not contemplated in its original definition. 17. Meaning of (-a).b, a.( 5b), and (-a).(-3d). 18. Meaning of (=) : (2) d n qg 19, 20. In what sense the Square Root of a Negative Quantity is impos- sible. 21-24. How we arrive at the meaning of the Sign —*. It may be called the Seminegative Sign. 25. Example of a Seminegative Quantity. 26. The square root of —Y* is the Seminegative of Y. 27. Meaning of the Signs = + in Co-ordinate Geometry. 28-30. Meaning of —, of Negative, and of Positive Quantities. vi TABLE OF CONTENTS. CHAPTER II. Nature of Co-ordinate Geometry. Representation of Points by means of Co-ordinates, Propositions respecting the Co-ordinates of Points. Re- presentation of Curves by means of Equations between 2 and y. ARTICLE 33. The Co-ordinates of a Point: what. 34. Origin of the name Co-ordinate, 35-37. Axes of x andy. Origin of Co-ordinates. Examples. 38. Relations between the Co-ordinates of two Points, the Line joining them, and the Angle that Line makes with OX. 39. Angle which one Line makes with another, what: distinction between the distances PP’ and P’P. 40-42. Areas of certain figures found. 43-46. Method of representing Lines and Curves by means of Equations between x and y. Loci. 47-49. Definition of the Locus represented by any Equation between x and y. Examples. CHAPTER III. General Equation of the Right Line. Equations of Lines subject to various conditions. Miscellaneous Propositions. Problems. ARTICLE 50. The general Equation of the first degree represents a Right Line. 51. Expression for the Angle which a Line makes with OX, 52, Converse of Art. 50. ' 53. The portions which a Line cuts off from the Axes found. 54, 55. Method of constructing a Line from its Equation. 55, 57 Of certain forms in which the Equation of the Right Line may be put. 58, 59, Current Co-ordinates and Parameters, what. Current Co-ordi- nates belonging to different Lines are not distinguished from each other. 60, 61. Equation of a Line passing through one or two given Points. 62, 63. Conditions necessary in order that two Lines may be parallel or perpendicular to each other. 64, 65. Point of Intersection of two Lines. 66. Angle contained between two Lines. 67. Equation of a Line making a given Angle with a given Line. 68. Distance of a Point from a Line in a given direction. 69, 70. Perpendicular distance of a Point from a Line. | 71. Equation of a Line passing through the intersection of two given Lines. 72-84. Various Problems respecting Right Lines. TABLE OF CONTENTS. vil CHAPTER IV. Oblique Co-ordinates. Modifications necessary in the previous results when the Co-ordinates are supposed to be Oblique. Polar Co-ordinates. Transformation of Co-ordinates. ARTICLE 85. Oblique and Rectangular Co-ordinates. 86. Articles which require no alteration when the Co-ordinates are supposed to be Oblique. 87. Alterations necessary in Article 38. 88, 89. Ditto in Article 50 and 51. 90,91. Ditto in Article 62. 92. Ditto in Article 66. 93. Ditto in Articles 68, 69. 94, Polar Co-ordinates, what. Radius Vector. Vectorial Angle. 95.96. Negative values of 7, how drawn. Examples. 97. Polar expression for the distance between two Points. 98,99. Polar Equation of a Right Line. 100. Transformation of Co-ordinates. 101. Transformation of Polar Co-ordinates into Rectangular, or vice versa. 102-104. To turn Rectangular Axes round the Origin through any Angle- 105-107. To transfer the Origin to any Point. 108. To reverse the positive direction of either Axis. 110. To turn Oblique Co-ordinates in any manner round the Origin. 112. To transfer the Pole and turn the Prime Radius through any Angle. 113-117. Various Problems respecting Lines referred to Oblique Co-ordi- nates, CHAPTER V. General Equation of the Circle. Diameters, Tangents, and Normals of a Circle. Various Problems respecting Circles and Right Lines. ARTICLE 118-121. General Equation of the Circle. 122, 123. To construct a Circle from its Equation. Examples. 124,125. Circle referred to Oblique Co-ordinates, 126, 127. Circle referred to Polar Co-ordinates. 128. Distance of a given Point from a Circle in a given direction determined by a Quadratic Equation. 129, 130. Sum and product of the Roots of the Equation. 131, 132. Distance of the middle Point of the Chord from the given Point. Condition necessary in order that the Point (ay) may be the middle Point of the Chord inclined at an Angle @ to OX, vill ARKTICLE 133, 134. 135, 136. 137. 138. 139. 140, 141. 142-164. TABLE OF CONTENTS. Locus of the middle Points of a System of Parallel Chords, what. Tangent at a Point determined. Tangent, making a given Angle with OX, determined. Equation of the Normal. Line joining the Points of Contact of the Tangents drawn from a given Point without a Circle. Locus of the Intersection of two Tangents when the Line join- ing the Points of Contact always passes through a Fixed Point. Various Problems respecting Right Lines and Circles. CHAPTER VI. Of the Loci called Conic Sections; their Forms and Equations. ARTICLE 165. 166. 167. 168. 169, 170. 171-175. 176. 177. 178. 180. 181, 182. 183, 184. Locus of a Point whose distance from a given Point is always proportional to its perpendicular distance from a given Line. Form of the Locus when e is < 1. Form of the Locus when e= 1. Form of the Locus when e¢ is > 1. Ellipse, Parabola, and Hyperbola. Origin of these names. Focus, Directrix, Centre, Axis, Major and Minor Axes in the Ellipse, Possible and Impossible Axes in the Hyperbola. Letters and Notation we shall use. Polar Equation of the three Curves. Latus Rectum equal to Double Ordinate through Focus. Locus represented by 4a* + By’? =C. This equation may represent two Right Lines. Locus represented by 4y’ = Bx or Az’ = By. CHAPTER VII. Of the General Equation of the second degree between z and y, and the Loci it represents. Sections of a Right Cone by a Plane are Loci of the second order. form. ARTICLE 185. 186. 187. 190. Reduction of the General Equation to its simplest General Equation of the second degree. Locus represented by the General Equation when the term 2Bzxy is wanting, how determined. The term 2Bxy may be made to disappear from the General Equation by turning the Axes of Co-ordinates through a certain Angle. Equations for reducing the general Equation so as to make 2Bay disappear. TABLE OF CONTENTS. ix ARTICLE 191-194. Examples of the reduction of the General Equation. 195. |The General Equation represents an Ellipse, Parabola, or Hy- perbola, according as 4 C’— B” is positive, zero, or negative. 196. Determination of 0, 4’, C’, D’, EZ’, in general. 197. Sections of a Right Cone by a Plane are Loci of the second order. 198. | Under what circumstances the section is an Ellipse, Parabola, or Hyperbola. CHAPTER VIII. Of the Parabola. Properties of the Parabola not connected with Tan- gents or Diameters. Properties connected with Tangents. Properties connected with Diameters. Various Problems. ARTICLE 201. Equations of the Parabola. 202. SP=x+m. 203. The distance of (zy) from S is a Linear Function of z and y. 204, Locus of a point (xy), whose distance from a given Point (hk) is a Linear Function of z and y. 205. Focus defined by this Property. 1 hie ae 207. SP fe SP = 7. 208-209. Line joining the Points when two Lines drawn from the Vertex meet the Parabola. 210. Distance of a given Point from a Parabola, measured in a given direction, is determined by a Quadratic Equation. 211. Sum and Product of the Roots of this Equation. 212. Condition that (kA) may be the middle Point of a Chord, making an Angle 6 with the Axis. 213. General property respecting the Products of the Segments of two intersecting Chords. 214, Equation of the Tangent at any Point. 215. Condition necessary in order that the Line (y = ax + 8) may be a Tangent. 216. Subtangent and Subnormal. 217-218. The Tangent makes equal Angles with 4<.X and SP. 219. Intersection of a Tangent, and the perpendicular upon it from the Focus. 220. Geometrical method of drawing a Tangent from any given point to the Parabola. 222. Analytical Solution of the same Problem. 223. Intersection of two Tangents at right Angles to each other. 224. ‘The perpendicular of SP drawn from S, meets the Tangent at P in the directrix. ¥ x ARTICLE 225. 226-222. 230. 231, 232. 233. 238-252. Of the Ellipse. Of the Eccentric Angle. ties connected with Diameters. ARTICLE 253. 254. 255. 256. 257-258. 259. 260. 261. 262-263. 265. 266-667. 268-271. TABLE OF CONTENTS. Length of the perpendicular from S on the Tangent. Line joining the Points of Contact of two Tangents drawn from a given Point. Equation of Diameter. The Tangent at the Vertex of a Diameter is parallel to its Chords. Equation of the parabola referred to any diameter and the Tan- gent at its Vertex. Equation of Tangent referred to the same Axes. Various Problems. CHAPTER IX. Properties not connected with Tangents or Diameters. Properties connected with Tangents. Proper- Various Problems. Equations of the Ellipse. SL er NS tS an — er, SP + S'P = 2a. Mechanical method of tracing an Ellipse. Distances of (xy) from the Focus, always a Linear Function of x and y. pete SPs a ae A Circle being described on the Axis Major as Diameter, Ordi- nate of Ellipse: Ordinate of Circle :: 0: a, Distance of a given Point from an Ellipse, measured in a given ‘direction, is determined by a Quadratic Equation, Condition that (hk) may be the middle point of a Chord making an Angle ® with the Axes of 2. General property respecting the Products of the Segments of two Intersecting Chords. Polar Equation of the Ellipse referred to the Centre. Equation of Ellipse may be expressed in the form 2 =a cos 9, y=bsin ¢. Geometrical Interpretations of this. Mechanical method of drawing Ellipses. Eccentric Angle: what. Tangent at any Point of an Ellipse found. a \ 5? Ce int Cle Z- Condition that (aa + By = y) may be a Tangent to an Ellipse. 2 « COM ENG Cae 3 ek. PG bisects the Angle SPS’. Normal found. ARTICLE 282. 283. 289-291. 292. 293. 295 256. 299. 300. 301. 303, 304-308. 309-319. TABLE OF CONTENTS. xi Locus of Intersection of any Tangent, and a Perpendicular upon it from either Focus. Geometrical method of drawing a Tangent from a given Point to an Ellipse. Analytical Method. Locus of Intersection of two Tangents at right Angles. pp = &. [ene | Ae p= sass Expression from the Perpendicular upon the Tangent from the Centre. Line joining the Points of Contact of two Tangents drawn from a given Point. Diameter of a given System of Chords determined. Tangent at extremity of Diameter is parallel to Chords. . b? tan 9 tan 6’ =-— —. a Conjugate Diameters: what. If @ and ¢’ be the Eccentric Angles at the extremities of two Conjugate Diameters, ?’ — ? = 90°. , a nee CA aa OM fd ae CP? + CD’, and CP.CD sin PCD are invariable. SPOS P= CD". Equation of the Ellipse referred to Conjugate Diameters, Equa- tions of Tangent, &c. &c. Various Problems. CHAPTER X. Of the Hyperbola. Properties of the Hyperbola corresponding to those of the Ellipse. Asymptotes. Problems. ARTICLE 320-326. 327. 328-355. 356-357. 358. 360-362. 363. 365-367. 368. 369-376. Properties of the Hyperbola corresponding to those of the Ellipse. The Conjugate Hyperbola : what. Properties of the Hyperbola corresponding to those of the Ellipse. Asymptotes of the Hyperbola determined. Equilateral Hyper- bola: what. Ellipse and Parabola have no Asymptotes. RQ = RQ, &e. &e. Equation of Hyperbola referred to its Asymptotes. Tangent referred to the Asymptotes. pp’ is constant. Various Problems. xu TABLE OF CONTENTS. CHAPTER’ XI. Miscellaneous Propositions. Vroperties of Conjugate Diameters deduced without using the Eccentric Angle. Locus represented by Ax’ + 2Bry + Cy’ = 1, determined by means of a Circle. Angle which a Tangent drawn from any point (ay) to a Conic Section subtends at either focus. Equal Conju- Areas of the three Conic Sections. gate Axes, &c. CHAPTER XII. General Equation of the Second Degree, Centre, Diameters. Tangents. Asymptotes. Various Problems, ERRATA. Page 5, third line of Art. 13, instead of (X -Y)+Y=0, and (Y- X)+X=0, read (X-Y)+Y=X, and (Y-X)+X=/Y. Page 54, last two lines of Art. 141, instead of and in the third inside it, read and in the third it may be inside it. Page 97, Art. 232, properly speaking, instead of saying that =4m', and TA=h; PR= PR, we should have said that PR=-PR, ~=— =-4m, and TA=-Ah. In fact we have considered PR’ and 7A without regard to sign. Page 123. The same remark applies to Art. 3804; we should have said that CP=- CP’, MP’=-(a+2x), CD=- CD, and MQ=- MQ. PLANE CO-ORDINATE GEOMETRY. CHAPTER I. MEANING OF THE SIGNS = +-. NATURE OF NEGATIVE QUANTITIES. MULTIPLICATION BY A NEGATIVE QUANTITY. - OF IMPOSSIBLE QUAN- TITIES, AND OF THE SIGN ae SENSE IN WHICH THE SIGNS = + -— ARE USED IN THE PRESENT TREATISE. Meaning of the signs = + -. Before we enter directly upon the subject of the present treatise, it will be necessary to make a few observations respect- ing some of the principles of Algebra; chiefly with a view tc explain the nature of negative quantities, and the use of the sign - in Co-ordinate Geometry. We shall commence with the following definitions. 1. The notation, X = Y, is a short way of expressing. the proposition, X zs equivalent to Y ; the sign = being an abbreviation for the words “7s equevalent to.” By the word “ equivalence” is meant ‘‘ sameness in certain particulars supposed to be understood.” ‘Thus, when we say that 42 shillings are equivalent to 2 guineas, we mean, that 42 shillings are the same as 2 guineas in a certain particular supposed to be understood, namely, in value; but not in weight or magnitude. Hence the notation X = Y may have a variety of significations according to the sense in which we use the word equivalent. Of course in every investigation the precise meaning of this word is supposed to be settled and understood. 2 PLANE CO-ORDINATE GEOMETRY. 2. The notation, X + Y, is a short way of expressing the words, X together with Y: the sign + being an abbreviation for the words ‘‘ together with.” 3. ‘The notation, X - Y, is used to represent that thing, whatever it be, which together with Y is equivalent to X: or, in symbolical language, (X -Y)+Y= X. The words “together with’’ are very general and may be used in a variety of different senses : therefore the sign + may have various meanings; and the same is true of the sign -, which, by the definition just given, has a signification depending upon that of +. The precise meaning of these signs, as in the case of =, is supposed to be settled and understood in every investigation. 4. The following examples will shew the nature of these signs thus defined, and some of the different senses in which they may be used, 5.328 —)B, Ge aed. eee ae (1), Sir Bies2) kote d ly 1 ee (2), £5 loss -++'£3 gain = £2\loss. 1), tare £5 loss — £3 gain = £8 loss .......... (4), £5 loss - £8 loss = £3 gain ........ (5), 8 miles travelled eastward + 2 westward = 1 eastward... .(6), 3 miles travelled eastward — 2 westward = 5 eastward. ...(7); 3 miles travelled eastward — 5 eastward = 2 westward... .(8), Ll OL CWE & 208.54 LIDIDEOI PS ee (9). Equation (1) translated into ordinary language may be ex- pressed thus: 5 together with 8 is equivalent to 8; therefore + here denotes common addition, and = denotes absolute sameness. Using + and = in this sense, we have, 2 + 3= 5; therefore, by the definition of X —Y, 2 is the value of 5 — 3, and the truth of the equation (2) is manifest. It is clear that — in this case denotes ordinary subtraction. In equation (3) + denotes that the loss and gain affect the same property, and = denotes sameness so far as the effect pro- duced on that property is concerned; and thus the equation expressed in ordinary language means, that a man’s losing £5 and gaining £38 is the same thing as his losing £2 so far as his property is concerned. PLANE CO-ORDINATE GEOMETRY. 3 Using + and = in this sense, the truth of the equations (4) and (5) is evident from the definition of X — Y; for £8 loss + £3 gain = £5 loss, therefore, by the definition, (4) is true; also £3 gain + £8 loss = £5 loss, and therefore, by the definition, (5) is true. In equation (6) + denotes that the traveller commences the 2 miles westward as soon as he gets to the end of the 8 miles eastward, and = denotes sameness so far as his final distance from his starting point is concerned: and thus in ordinary language the equation means, that a man’s travelling 3 miles eastward and then 2 miles westward is the same thing as his travelling 1 mile eastward, so far as his final distance from his starting point is concerned. Using + and = in this sense, the truth of the equations (7) and (8) follows immediately from the definition of X — Y, as in the case of (4) and (5). In equation (9) + has merely the force of the particle and, and = signifies that the things on the first side of the equation are respectively or separately equivalent to those on the second side ; and thus in ordinary language the equation means, that £1 and 1cwt. are respectively equivalent to 20s. and 112 lbs. ‘This explains the meaning of such an equation as a+6W-1l)=c+dv- 1). 5. It is evident that 5+ 3, £5 loss + £3 gain, and 38 miles eastward + 2 westward are the same things, respectively, as 3 +5, £3 gain + £5 loss, and 2 miles westward + 3 eastward; and in general that X + Y is the same thing as Y+ X. At least, we shall never attach any meaning to the words “ together with” for which it is not true that X together with Y is the same thing as Y together with X. 6. The 5th and 8th of the above examples serve an important purpose ; they shew that the definition we have given of X ~Y applies, without any difficulty whatever, when X is a magni- ude of the same kind as, and numerically less than, Y; and that we are therefore at liberty to suppose X to be of any magnitude we please compared with Y. This observation is important, as we we now about to use a notation which would be inadmissible, if B2 4 PLANE CO-ORDINATE GEOMETRY. there were any restriction implied in the definition of X - Y requiring X to be a magnitude numerically greater than Y. Of Negative Quantities. In various parts of mathematics we often meet with two quan- tities of such a nature, that, when taken together, they are equivalent to zero. Thus £1 loss and £1 gain, taken together, are equivalent to zero (using the words “taken together” and “ equivalent” in the same sense as in Ex. (3) Art. 4). Again, using these words as in Ex. (6), 2 miles travelled eastward and 2 westward, taken together, are equivalent to zero. And a variety of other instances might be given, such as, two equal and opposite forces or velocities, vitreous and resinous elec- tricities, &c. Hence, in order to distinguish and represent two quantities thus related, it becomes desirable to adopt a special nomenclature and notation, which we now proceed to explain. 7. When two things taken together are equivalent to zero, each is said to be the negative of the other; or, in symbolical language, if X + Y= 0, X is said to be the negative of Y, and Y the negative of X. 8. It is usual to select either of the two things thus related and call it posetive, and then the other is said to be negative: thus we may call gain positive and loss negative; or, if we please, we may call loss positive and gain negative. We use the term positive, therefore, not to express any peculiarity in the nature of a quantity (except it be that it is capable of having a negative), but merely to mark, for convenience sake, one of the two quantities which taken together are equivalent to zero. 9. We arrive at a very excellent notation for representing the negative of a quantity in the following manner. By the definition of X - Y above given, it appears that 0 —- ¥ is that quantity which together with Y is equivalent to zero; hence, by the definition of a negative, 0 — Y is the negative of Y, Now, in such expressions as 0 + Y and 0 -Y, it is usual to omit the symbol 0, and to write them simply in the form +Y -Y, and thus —Y, instead of 0 —Y, is employed to represent the PLANE CO-ORDINATE GEOMETRY. a negative of Y. In this manner it is that our definition of X — Y leads to a method of denoting the negative of a quantity (see the observation made in Art. 6), and to the use of the sign — as an independent sign. We may regard —in this point of view as an abbreviation for the words “ the negative of”; so that - Y expressed in ordinary language means, the negative of Y. 10. 0+Y is evidently the same thing as Y, and therefore, if we omit the symbol 0, + Y and Y mean the same thing. There does not however seem to be any advantage gained by making this independent use of the sign +, except in a few cases where it avoids circumlocution, such as the common notation + V(X). 11. If we bear in mind that —-Y is merely a simple way of writing 0 -Y, which, according to our definition, denotes that quantity which together with Y is equivalent to zero, there will be no difficulty in perceiving the truth of the various proposi- tions respecting the sign — usually given in treatises on Algebra; for instance, the following : 12. ‘To shew that -(-Y)= Y. Since (0 -Y) + Y= 0, by our definition of X — Y, it follows, from the definition of a negative, that Y is the negative of (0 - Y); and therefore that Y= 0-(0-Y); or, omitting the ‘zeros, that Y=-(-Y). Q.E.D. 13. To shew that -(X -Y)=Y- X. By our definition of X - Y, we have (X-Y)+Y=0, and(Y- X)+ X=0; ‘therefore, adding these equations, and taking away from each side of the result the common quantity X +Y, we find (X-Y)+(Y- X)=0. Hence, by the definition of a negative, we find that (Y - X) is the negative of (X —Y), and therefore Y-X=0-(X -Y)=-(X-Y). Q.E.D. 14. To shew that, to add - Y to any quantity X, is the same as, to subtract Y, and, to subtract — Y, is the same as, to add Y. Since (0 -Y)+Y = 0, it follows that X+(0-Y)+YV=X; 6 PLANE CO-ORDINATE GEOMETRY. and therefore, by the definition of X — Y, we have X+(0-Y)=X-Y, which shews that X together with - Y is the same thing as oY SQ, Ev. Again, since X + Y+(0—Y)= X, we have, by the defini- tion of X ~ Y, BG Pere Cds (Ra et which shews that to subtract — Y is the same thing as to add Y. Q. E. D. Of Multiplication by a Negative Quantity. 15. To determine the meaning of multiplication by a nega- tive quantity we must proceed very much in the same manner we do in the case of negative or fractional powers. The original definition of a” necessarily supposes that m is a positive integer ; but, in order to extend the meaning of this notation, we assume, that all the consequences which follow from the original defini- tion, and which may hold equally well whether m be an integer or not, are true in general. Thus the law, aa" = a”, follows from the original definition, and is capable of holding when m and m are not integers: we therefore. assume it to be true in general, and it determines the meaning of a” when m is negative or fractional. 16. In exactly the same way, the definition of a. originally given supposes the factors to be integers; but from this definition it follows that (a+ c¢).6=a.64+c.b and that @.(6+¢)=a.b+a.c; which laws are capable of holding when a, 4, and c are not integers. We therefore assume these laws to be true in general, and they serve to determine the meaning of a product when the factors are negative or fractional, as we now proceed to shew. 17. To determine the meaning of (- a).6, a.(— 6), and (- a).(- 3d). By the assumed laws we have {0-a)+a}.b=(0-a)bia.h; but (0 -a@)+a@=0, and 0 - @ is the same thing as — a; therefore, since 0.6 = 0,* we have 0=(—a).b6+a.b,and..(—a).b=-—(a.b). * That, 0.5 = 0, follows from the original definition of a. 6. PLANE CO-ORDINATE GEOMETRY. ty Hence it appears that the result of multiplying 6 by - a is the negative of a. 5b. In just the same way we may shew that a.(- 6) = -(a.6). Also, by the results just obtained, we have (- a) (- 6) =- {a.(- 6) =- Ca.b)=a.), by Art. 12. Hence the product of —- a and - disa.d. 18. To determine the meaning of (=) : (2) A n From the assumed law, (a+ ¢).b=a.b+c.), it easily follows that, (a@+c+die+&c...)b=a.b+c.b64+¢.b64+d.b4+&e.. therefore we have (Se ee. .-.tonterms) 2-7 24% 2. ton ters, ae Ro ON A ag a! eR or m Pan(@.2) Tote rate Oe GL q n 9g and in exactly the same way we may shew that woe (mt)orn (Ee )n0 Therefore, dividing by ¢., we have m p_m.p Rig n.g Hence the product of two fractions is the faction, whose numerator is the product of the numerators, and denominator the product of the denominators. Of Impossible Quantities, and the Sign —* or V(—1). 19. It appears from Art. 17 that there is no positive nor negative quantity whose square is a negative quantity: the square root of a negative quantity, therefore, is neither positive nor negative. Now suppose that, in any investigation, all the quantities we are engaged with must be either positive or negative, at least, that we agree to consider no other kind of quantity : then, if any of our operations should lead to a result in the form of /(- a), we must reject this result, and regard the operations that lead to it as impossible to be performed. In 8 PLANE CO-ORDINATE GEOMETRY. this point of view the square root of a negative quantity may be strictly called impossible or imaginary. 20. In the first part of the present treatise we shall consider all the quantities we are engaged with to be either positive or negative, because there will be no necessity to introduce any other kind of quantity; on the contrary, the simplicity of the subject would be quite lost, and no advantage would be gained, if we were to admit any quantities besides simply positive and negative quantities. This being settled, we must reject the square root of a negative quantity whenever it occurs, and regard the operations that lead to it as impossible. 21. But the square root of anegative quantity is impossible only in this point of view, for we shall now shew that there is a quantity whose square is a negative quantity. 22. -—Y represents the result of a certain operation performed upon Y, namely, the operation of converting a quantity into its negative ; —(— Y) denotes the result of performing this operation upon — Y, @.e. twice successively upon Y; -{—(-Y)} the re- sult of performing it three times successively upon Y, and so on. To represent these results of repeating the operation more concisely and generally, we shall denote them respectively by —Y, — Y, and so on; and, in general, we shall assume —-” Y to denote the result of performing the operation m times succes- sively upon Y. In like manner, if we suppose + Y to denote the result of any other operation of any kind performed upon Y, we shall assume +”Y to denote the result of performing it m times successively upon Y. 23. This definition of +” Y supposes m to be an integer, but we may extend the meaning of the notation, in the same manner as we do in the case of ordinary powers, by supposing that all the properties which follow from the original definition, and are capable of holding when m is not integral, are true in all cases, whatever m may be. For instance, it follows from the original definition that + (+ ” Y) =-++ ™" Y, and this property is capable of holding when m and ” are not integral; we shall therefore assume it to be true in general. PLANE CO-ORDINATE GEOMETRY. 9 Hence, to determine the meaning +? Y we have only to put m = n= in the assumed property, and we find a 1 L bread )ae—te which shews that +-% represents any operation which, performed twice successively on any quantity, gives the same result as the operation + performed once. 24. Hence -2 represents any operation which, repeated twice successively upon a quantity, converts it into its negative. On this account I shall venture to call -? the seminegative* sign, and -: Y the seminegative of Y. Hence the seminegative of the seminegative of Y is the negative of Y; or, in other words, the same operation which converts Y into its seminegative, con- verts the seminegative of Y into the negative of Y. 25. For example, by turning a traveller’s direction through a right angle from right towards left, a mile eastward is changed into a mile northward; and, by exactly the same change, a mile northward is changed into a mile westward ; but a mile west- ward is, as we have seen, the negative of a mile eastward; hence it follows that a mile northward is the seminegative of a mile eastward, and if we represent the former by Y the latter is re- presented by -? Y. 26. We have now obtained a quantity whose square is a negative quantity, for it is easy to shew that (-? Y).(-? Y)=-Y° in the following manner. It follows, from the original definition of —” Y, and from the nature of multiplication by a negative quantity explained in Art. 17, that (-"X).(-"X) = —-""(X.Y); and this law is capa- ble of holding when m and x are not integral; we may therefore * The sign 3 or 4/(—1), as it is commonly written, seems to me to require some specific name to indicate the reality of the operation it represents. Be- sides, since : 7 1 Ls 0 (— 1)” = cos >, Sin /(- 1), all the roots of —- may be expressed very simply by means of its square root ; and therefore the square root may be advantageously distinguished from the other roots:by a special name. 10 PLANE CO-ORDINATE GEOMETRY. assume it to be true in general. Now, putting m =n = 3 and X =Y, we find immediately, (-2 Y).(-? Y)=-Y". Q. E. D. Hence it appears that the square of the seminegative of Y is the negative of Y. (See Art. 21.) We shall return to the consideration of seminegative quanti- ties in the second part of this treatise. Meaning of the Signs = +—- in Co-ordinate Geometry. 27. In the following pages we shall generally employ the slgns = + — in the same manner as in the examples of distances performed by a traveller eastward or westward, given in Art. 4, Ex. (6), (7), (8); for we shall, for the most part, be engaged with the consideration of similar distances, which are supposed to be traced by the motion of a point in particular directions. Let A, a, a’, a’, &c. represent any distances described along a right line AB (fig. 1) by a tracing point moving either from right to left or from left to right ; and, to avoid circumlocution, let us call the position of the tracing point, when it commences describing any distance, the beginning of that distance; and its position when it has completed it, the end of the distance. Then, by the equation ata+aia’+&e.= A, we shall always mean, that the end of the distance @ is the beginning of a’, the end of a’ the beginning a’, the end of a’ the beginning of a’, and so on; and that when the tracing point has described the compound distance a + a’ + a’ + &c. (beginning a at the end of a, a’ at the end of a’, and so on), its ultimate position is the same as if it had simply described the distance A; or, in other words, that the beginning and end of the compound distance a + a + a’ + &c. coincide respectively with those of A. 28. Having thus settled the meaning of the signs = and +, that of the sign - may be determined in the following manner. Let a and 6 represent two distances equal in magnitude, the former being described from left to right, the latter from right to left; then, using = and + in the sense just settled, it is evident PLANE CO-ORDINATE GEOMETRY. ll that 6+ a= 0; therefore, by the definition of a negative, 6 is the negative of a, and is represented by 0 — 6 or — 6, as we have shewn (see Art. 9). Hence it appears that if a represent any distance described from left to right, —- @ denotes a distance equal to it in magnitude, but described from right to left; and the reverse is also true. 29. Since a-b=a+(—)) (see Art. 14), it follows that a-6 represents the final distance of the tracing point from its original position, when it has described the distance a, and then the distance (— 6), which latter denotes a distance measured equal to 6 in magnitude, but opposite to it in direction. And this is true whatever distances a and 6 may represent. 30. It appears hence, from Art. 8, where we have explained the meaning of the term positive, that if we regard distances measured from left to right as positive, those measured from right to left will be negative ; or, if we choose to call the latter distances positive, the former will be negative. 31. Allthat we have said respecting distances described from right to left or from left to right applies equally well to distances described along any line in any position, whether it be a right line or a curved line. The same may also be said of angular distances described by a revolving line. These few observations respecting the signs = + and - suffice for our present purpose, and we shall now proceed to explain the nature and object of Co-ordinate Geometry. CHAPTER II. NATURE OF CO-ORDINATE GEOMETRY. REPRESENTATION OF POINTS BY MEANS OF CO-ORDINATES. PROPOSITIONS RESPECTING THE CO-ORDINATES OF POINTS. REPRESENTATION OF CURVES BY MEANS OF EQUATIONS BETWEEN & AND ¥Y. 82. By Co-ordinate Geometry we mean that method or system invented by Descartes, in which the positions of points are determined, and the forms of curves and surfaces defined and classified, by means of what are called co-ordinates. In the present treatise we shall confine our attention to one plane, and it is on this account that we have given the name of Plane Co-ordinate Geometry to the subject. We shall suppose that all the points, lines, &c. we have occasion to consider, lie in the plane of the paper, except in certain cases, where it will be necessary to make use of solid figures. Method of representing the Position of a Point by means of Co-ordinates. 33. Let OX, OY (fig. 2) be two right lines perpendicular to each other, and P any point: draw PM perpendicular to © OX and PN to OY. Then OM and OW are called the co-ordinates of the point P with reference to the lines OX, OY, which are termed the co-ordinate axes. It is clear that these co-ordinates serve to determine the position of P with respect to OX and OY; for if OM and OW be given, the lines MP and NP are also given, and therefore their point of inter- section, which is P, is known in position with respect to OX and OY. 34. The lines OM and OW acquired the name of co- ordinates in the following manner. From any series of points P, P’, P", &c. (fig. 3), suppose that we drop perpendiculars PLANE CO-ORDINATE GEOMETRY. 13 PM, PM’, P’M", &c. upon a given right line OX; then, if these perpendiculars, and the portions OM, OM', OM", &c. which they cut off from the given line, be given in order and magnitude, the points P, P’, P’, &c. are determined in order and position. On this account the perpendiculars were called ordinates (from ordino, which signifies to arrange in order or succession), and the portions OM, OM’, &c. cut off from OX were called the abscisse of the ordinates. Now, if we draw OY at right angles to OX, and drop the perpendiculars PJ, P'N', P'N" upon it, these perpendiculars, which are the ordinates of the points with respect to OY, are respectively equal to the abscisse, and may be used in place of them: and thus the ordination, so to speak, of a series of points, may be effected by a double system of ordinates instead of a system of ordinates and abscisse. In this point of view the lines PM and PN, or OM and ON if we please, are naturally called the co-ordinates of the point P. Thus it was that the lines OM and ON came to be designated by the term co-ordinate. ‘The ordinates of a series of points were called by Newton “ lhinee ordinatim applicate,” and by some authors the ordinate of a point was termed “crus efficiens,” and the abscissa “ crus patiens :” but the name co-ordinate is now universally used. 39. Any distance described from O along OX is generally denoted by the letter z, and any distance along OY by the letter y. On this account OX is called the axis of x, and OY the axis of y. ‘The point O, where the two axes meet, is called the origin. All distances described to the right along OX, or upwards along OY, we shall consider to be positive; and then all distances described to the left along OX, or downwards along OY, will be negative. The principles upon which we do this are explained in Arts. 27-31. We shall call OX the positive axis of z, OX’ the negative axis of z, OY the positive axis of y, and OY’ the negative axis of y. The point whose co-ordinates are x and y we shall term, the point (xy), to avoid circumlocution. 36. Hence if a and b be any two positive distances, 7. e. dis- tances described by a tracing point moving along OX or OY in 14 PLANE CO-ORDINATE GEOMETRY. the positive direction, and if «=a and y = 3, the point (zy) 1s determined by describing OM and OWN equal in magnitude to a and 4, as in (fig. 4), and then the intersection (P) of the perpen- diculars MP and NP will be the point (zy). If z =- aand y = 4, the distance OM must be described in the negative direction, by Art. 28, and therefore (fig. 5) repre- sents the proper position of P. If z=aand y=- 6, ON must be described in the negative direction, and therefore (fig. 6) represents the position of P. If z=-—aand y = — 6, both OM and ON must be described in the negative direction, therefore (fig. 7) represents the posi- tion of P. We may perceive, hence, the important use of negative quantities in the present subject; without them it would be necessary to assume one set of letters to represent distances described in the positive directions, and another set to denote those described in the negative directions. 37. The following propositions will serve to illustrate this method of employing co-ordinates to represent the positions of points: the first of them is of considerable importance, and we shall often have occasion to refer to it. Prop. I. 38. P and P’ being any two points referred to co-ordinate axes OX, OY (fig. 8), to determine what relations subsist between the co-ordinates of P and P’, the distance PP’, and the angle which PP’ makes with OX. Let zy and 2'y’ be the co-ordinates of P and P’, draw PM, P’M' perpendicular to OX, and PQ parallel to OX: then PQ= MM =0M'-OM=2-2z, PQ=PM-PM=y'-y, and 2 P'PQ=96. Hence, from the right-angled triangle PP’Q, we have ' z-—-zx=receos, y'-y=rsin 0,). P= (2-2) 4 (y'- J, PAE tan 6 = y-y ae — a” which are the relations required. The two latter are immediately deducible from the two former. PLANE CO-ORDINATE GEOMETRY. 15 39. It is important to observe that by “the angle which PP’ makes with OX,” we mean the angle through which OX must be turned towards OY, before it becomes parallel to PP’, with the point X on the same side of O that P’ is of P. Thus, in (figs. 9 and 10), @ is the angle which PP’ makes with OX, and @ the angle which P’P makes with OX. Also by the distance PP’ we mean the distance measured from P to P’, and not from P’ to P. These remarks are made to avoid ambiguity in speaking of angles and distances, and will be found particularly important when we come to consider distances measured along revolving lines. Prop. II. 40. To determine the area of the trapezium PMM'P’ (same figure). Let A be the required area; then 2A = 2 area PMM'Q + 2 area PQP’ = 2y (@ — 2) + (y— y) (@ - #) =(y¥ + ¥) @ - 2); which gives the required area. ed side A bb 41. To find the area of the triangle whose angular points are (xy), (vy'), (y’). Let PP’P" (fig. 11) be the triangle, draw PM, P'M', P"M" perpendicular to OX ; then, if A be the required area, we have A= PMM'P'+ P?M'M'P"- PMM'P". Hence, by the preceding proposition, 2A =i +y)@-2)+Yry)@-xH)-Y'+y @ - 2), or 2d =(y'+y)(@-z)+yty)@-2)+¥+y)@- ©), which gives the required area ina remarkably symmetrical form. 42. Cor. In exactly the same way we may shew that twice the area of any polygon, whose angular points are (zy), (zy’), (z'y’). ...(a@™y™), is equal to (y'+ y) (@'-2) + (y+y) (2-2) + (yt) (@"-@)net (yt y) (2-2). By this formula we may easily determine the area of a field 16 PLANE CO-ORDINATE GEOMETRY. whose boundaries have been determined by means of what are called offsets in surveying. Method of representing Curves by Equations. 43. We shall now explain the manner in which co-ordinates may be applied to determine the forms of curves ; it is in this application of them that Co-ordinate Geometry chiefly consists. Let z and y be the co-ordinates of any point P, and suppose that there is given, not z and y, but only some relation between them ; then we may assign any value we please to z, and, by means of the given relation between z and y, find a correspond- ing value of y. If we give to z any set of values OM, OM’, OM", &c. (fig. 12), and find the corresponding values of y, MP, M'P', M"P", &c. suppose, the point we are considering may be in any of the positions P, P’, P”, &c.. ...; and, since the values we give to x may differ from each other as little as we please, it is clear that there is, 7 general, some continuous line or curve in which all the positions of the point will be found. We say, 7 general, because in certain cases the point may occupy only isolated positions, as we shall presently shew. We have supposed that there is only one value of y for each value of z, but the given relation may furnish more than one value; in such a case all that we have just said still holds true, only the curve in which the various positions of P are found will consist of | more than one branch, as in (fig. 13). 44. Hence it appears that, when there is given not z and y, but only some relation between z and y, the position of the point (zy) is indeterminate, but is so far restricted as to be always found upon some line or curve (in general), the form of which depends, of course, upon the nature of the given relation. 45. When an indeterminate point is restricted by conditions of any kind to occupy some one of a particular series of posi- tions, that series of positions is called the Jocus of the point. A relation, therefore, of any kind between z and y represents in general some locus, namely, that series of positions which the point (zy) may occupy consistently with the given relation. PLANE CO-ORDINATE GEOMETRY. Re 46. In the following pages we shall only consider those relations between x and y which may be expressed by ordinary equations (for there are relations which cannot be expressed by means of ordinary equations); and therefore every locus, we shall be concerned with, will be represented by some equation between x and y, by means of which we shall investigate its nature and properties. 47. We therefore define the locus, represented by an equa- tion between zx and y, to be the assemblage of all the points whose co-ordinates satisfy that equation. 48. Hence if 4 and & be quantities which, substituted for x and y, satisfy the equation, (A) must be a point of the locus. 49. The following examples will shew the nature of this method of representing loci by means of equations. To determine the nature of the locus represented by the equation y = x ~ a, a being some known positive distance. Take OM (fig. 14) as any value of xz, let OA =a, draw MP perpendicular to OX, and take MP = MA ; then, since MP = OM- OA, MP is the value of y corresponding to the value OM of x; and therefore P is a point of the locus to be determined. Now, since MP = MA, the angle PAM = 45°; therefore P '$ a point on the right line drawn through -A at an angle 45° to she axis of x; and this is true whatever value of x we suppose OM to be. Therefore all the positions of the point (zy) are ound on this line ; and hence the equation (y = e - @) repre- ents a right line drawn at an angle 45° to the axis of x, and mutting it at a distance a from the origin. To determine the nature of the locus represented by the quation, z* — 2ax = 2ay — y’ - a’. This equation may evidently be put in the form (2 - a) + (y¥ — ay =a"... .(1). C 18 PLANE CO-ORDINATE GEOMETRY. Hence, if C and P (fig. 15) be the points (aa) and (zy), CP is equal to a whatever value we give to 2. Therefore the given equation represents a circle whose centre is C’ and radius a. Having thus explained the method of co-ordinates, we now proceed to apply it in detail to the investigation of the properties of right lines, circles, and various other curves, and the solution of several interesting and important problems. CHAPTER III. OF THE EQUATION OF THE RIGHT LINE. EQUATIONS OF RIGHT LINES SUBJECT TO VARIOUS CONDITIONS. MISCELLANEOUS PROPOSITIONS, PROBLEMS. Of the General Equation of a Right Line. Prop. IV. 50. To shew that the general equation of the first degree between z and y represents a right line.* Every equation of the first degree between z and y is included in the form Agee Bue Oca cg ae (1), A, Band C being any quantities independent of x and y; and this equation is therefore called the general equation of the first | degree. Let (2'y’), (x"y") be any two points of the locus, whatever it _be, represented by (1); then 2’, y’, and 2’, y" put for z and y ‘must satisfy (1) (Art. 47), and we have therefore Az'+ By' =C, Az’ + By" =C; and, subtracting these equations, we find A yo ad y ks Bibra gOy Now if 6 be the angle which the line joining (z'y’) and (zy) makes with the axis. of z, we have, by Prop. 1., tan 9 = Ee ee A Hence we find tan 9 = - 3. * Hence it is that an equation of the first degree is often called a linear equation. a2 20 PLANE CO-ORDINATE GEOMETRY. It appears therefore that the right line joining any two points of the locus represented by (1) always makes an invariable angle with the axis of z; which can only be true of a right line. Therefore (1) represents a right line. Q.£.D. 51. Cor. Hence — = is the tangent of the angle which the right line (Az + By = C)* makes with the axis of z; 7.e. the line OX must turn through an angle tan™ - 3 towards OY, in order to become parallel to the line (Av + By = C) (see Art. 39). Hence, if AB (in either fig. 16 or fig. 17) be the line represented by the equation, BAX, not BAO, is the angle whose tangent is- 5. Prop. V. 52. To determine, conversely, the general equation of a right line. Let AB (fig. 18) be any right line, P any point of it, and OM (= x), MP (= y), the co-ordinates of P: then, whatever be the position of P on AB, we have PM: MA in some invariable ratio, B:.A suppose; hence, putting O-A =a, we find & —o = =, or Az + By = Ba=C, suppose ; which equation, being a relation between the co-ordinates of any point of the right line, is the equation required. Prop. VI. 53. To determine the portions which the line (4z+ By=C) cuts off from the co-ordinate axes OX, OY. Let BA (fig. 19) be the lme (Ax + By =C), meeting OX and OY in A and B respectively ; then OA, OB are the portions the line cuts off from OX, OY. Now A is a point of the right line, and the co-ordinates of A are OA and 0; therefore the equation must be satisfied when we put z= OA, y= 0; “e. OA is the value of z got from the equation by putting y = 0. * By the line (4x + By=C) we mean the line represented by this equation. PLANE CO-ORDINATE GEOMETRY. WA Similarly OB is the value of y got by putting x= 0. Hence we have iy C es C = ae a B . Q. E. F. 54. Cor. Hence we may easily construct the right line represented by any given equation, as the following examples will shew, viz.: Bierat hale BOCs Ne Wal siete seh GL); 382-2y= 6a (2), Se ete tte OE tte wieiep cess «ic 0) abe (3), 8x + 2y = — ba ways L=a HGS) ae (6) In (1) we have x = 2a when y = 0, asacyek Scaler z= 0; therefore take OA = 2a, OB = 8a (fig. 20), and AB is the right line represented by (1). In (2), OA= 2a, OB = - 3a, therefore fig. 21 represents (2). In (8), OA =- 2a, OB= 8a, therefore fig. 22 represents (8). In (4), OA =- 2a, OB = — 26, therefore fig. 23 represents (4). (5) represents a series of points all at the same perpendicular distance from O Y, z.e. a right line parallel to OY (fig. 24): and similarly (6) represents a right line parallel to OX (fig. 25). (5) and (6) may be put in the forms z+0.y=a, 0O.r%+y=a4, from which it appears that OA = a, OB =m in the former, and OA =a, OB =a in the latter. 5d. There is one case to which this method cannot be applied, namely, when the equation occurs in the form Az + By =0, for in this case OA and OB are each zero, which shews that the line passes through the origin, but does not determine its position. But we may immediately find its position, since, by Prop. rv. Cor., the tangent of the angle (@) it makes with OX A is—-—,. For example, to construct the lines represented by B Sate Pe ia) oo cbc 4 vieic y kee ee OUEt Ore oate'e 8 ke eee CO), 22 PLANE CO-ORDINATE GEOMETRY. In (7), tan 6 =~ 1, and therefore 0 = ou , and fig. 26 represents the line. - In (8), tan 0 = 1, and therefore if we take OM = 3, MP = 1, (fig 27), the line drawn through O and FP is that represented by (8). 06. Of certain forms in which the equation of a right line may be put. 12 ANSI) amet (1) If we divide the equation by C, and put C5) Cee it assumes the form BMG Gi Seyi Here a and @ are the portions which the line cuts off from the axes OX, OY; for x =a when y = 0, and y = 6 when z= 0. (2) If we divide by Band put - = =, ¢ =c, the equation assumes the form Y = MZ +. Here m is the tangent of the angle which the line makes with the axis of x; and, since y = c when z = 0, ¢ is the portion cut off from the axis of y. If therefore we take OB = c, and draw the line AB making the angle BAX = tan”m (as in fig. (28) if m be positive, or as in fig. (29) if m be negative), AB is the line represented by the equation. | (3) If we put tan @ for m in the equation y = mz +c, multi- ply by cos 0, and put ¢ cos 0 = p, it assumes the form y cos @-2 sin 0=p. Here 9 is the angle which the line makes with the axis of z ; and, if we draw OQ perpendicular to AB, p= OQ; for p=ccos 8= OBcos BOQ = OQ. * This equation may be proved geometrically as follows. In fig. (18) we | Su) NP_BP MP _ AP. OAR BA 1°00 bee (xz pace ipa) HOA VOR wl hia ca gt 7 : + b 1. or PLANE CO-ORDINATE GEOMETRY. 93 57. Hence it appears that, when the equation of a right line is put in the form 2 sy 1, BP la a and # are the portions it cuts off from OX and OY. - When it is put in the form Y = ML + C, m is the tangent of the angle it makes with OX, and ¢ is the portion it cuts off from OY. And when it is put in the form y cos 9- x sin 0 =p, @ is the angle it makes with OX, and p is the perpendicular upon it from O. Of the Equations of Right Lines subject to various Conditions. 58. When we have occasion to write down the equations of several different lines at the same time, we shall not make any distinction between the 2’s and y’s in the different equations, but write the same z and y in each of them: which amounts to supposing that (zy) is any point on any of the lines. Thus, suppose we have to consider two lines ; then, instead of saying, let their equations be ee eA 2 ea, we shall simply say, let their equations be meet by = CL... CE epee eee Ys. Os rte crs (2): This being the case, it will be important to remember that (zy) in (1) denotes a point on the line represented by (1), and (zy) in (2) a point on the line represented by (2); and that therefore (zy) in (1) must be a different point from (zy) in (2), except at the intersection of the two lines. 59. The general co-ordinates of any point of a locus, between which the equation of that locus is a relation, have been called by some writers the current co-ordinates of the locus ; and the other quantities involved in the equation, which are independent of these co-ordinates, and serve to determine the position and form of the locus, are generally called the parameters of the locus. Thus, in the equation of the right line, 94 PLANE CO-ORDINATE GEOMETRY. x and y are the current co-ordinates, and A, B, C the para- meters. ‘These names are very convenient in many cases. Prop. VII. 60. To determine the form of the equation of a right line when it is restricted to pass through a point (AA), or through two points (A£) and (h’2’). Let the equation of the line be Az By = G55 Gale as OE e (1), Then, since (hk) is a point of it, we have Ah fi Bh AG aes ae eee (2% (2) determines one of the parameters A, B, C in terms of the other two, and expresses the condition necessary in order that (1) may pass through (hk). If we subtract (2) from (1), in order to get rid of C, we find ; A (e— hh) By — fh) = On. eee (3) ; and, since this equation is manifestly satisfied when xz =’ and y = k, it is the general equation of a right line restricted to pass through the point (hh). If the line also pass through the point (/’/’), (3) must be satisfied when x = h' and y = k'; therefore Ath —h)+ B(k —kh)= 00 aes (4), (4) is the condition necessary in order that the line (3) may pass through (/'k'); and, if we substitute in (8) the value of 4+ B got from (4), (3) becomes zt-h y-k rar pt Ons heep aaah eh (5), which is the equation of a right line passing through the two points (h/) and (h’h’). 61. Cor. Hence the equation of the line drawn through (Ak) at an angle @ to the axis of z is (z — h) sin 0 -(y— k) cos 0 = 0, as is evident by putting in (3) for = its value (— tan @). Prop. VIII. 62. ‘To find the conditions necessary in order that two right lines (4x + By = C) and (A’xz + By = C’) may be parallel or perpendicular to each other. PLANE CO-ORDINATE GEOMETRY. 25 If 0 and 0’ be the angles which these lines make with OX, we have, by Prop. tv. Cor., A A’ tae st a i ie an B an @ 7 Hence, if the lines be parallel and therefore 6’ = 0, we have AeA a (1). And if the lines be at right angles, and therefore 0’ =04 7 which makes tan 6’ = - cot 0, we have ¢ A! B RiP eL pte ee (2), (1) and (2) are the conditions required. 63. Cor.1. Hence the lines (Az +By=C) and (Az+By=C’) are parallel, and the lines (Az + By =C) and (Bz - Ay =C’) are perpendicular to each other. Cor. 2. Hence, and by Prop. vit., the equation of the lines drawn parallel and perpendicular a (Az + By = C), through the point (24), are respectively, A(x-h)+Biy-k)=0, B(z-h)- A(y-k)=0. Miscellaneous Propositions respecting Right Lines. Prop. IX. 64. To determine the point of intersection of two right lines, viz. Fo ERGY 8 EH EMER Roe. et Ae ay Pee Ue Onis scene, (2): As we have explained in Art. 58, z and y are not the same quantities in (1) that they are in (2), except they belong to a point common to both the lines: suppose therefore that 2 and y are the same in (1) and (2), and then ( ‘ay) 1s a point common to the two lines. Now, on this supposition, (1) B’ - (2) B and (1) A'- (2) A give e CB'- C'B _CA-C'A WARES aaa ip The values of z and y thus determined are the co-ordinates of the point of intersection required. 26 PLANE CO-ORDINATE GEOMETRY. 65. Cor. These values of x and y become infinite when AB' — A'B = 0, as they ought to do, for then, by Art. 62, the two lines are parallel. PROPS 66. To find the angle which the line (4'z + B’y = C’) makes with the line (Az + By =C). Let ¢ be the required angle, and 6’, 0 the angles which the two lines make with the axis of z; then we have / o¢=6-6, tan 0 =- ae tan 0 =- rae B B ALA BAe d theref t = : an ererore an = a ry Bee AB'-A'B OY tan p = —__,—____. , AA + BB' | which determines ¢. ra ee V{(A? + B’) (A? + Bt Prop. XI. | 67. To find the equation of a line making a given angle (@) with the line (Az + By = C). The angle which the required line makes with the axis of z Cor. Hence cos ¢ = A ; is p + tan” (- 5) and its tangent is therefore CADP Tee - Bsin $ - A cos ¢ ; Boos ¢+ Asin po 1 + tan cw) B Hence the equation of the required line is, by Art. 51, (A cos ¢- Bsin ¢)x+(Asing + Boos o)y=C. Prop. XII. 68. ‘To find the length (7) of a right line drawn, at an angle § to OX, from a point (AA) to a line (Az + By = C).. Let us suppose (zy) to be the point where the former line PLANE CO-ORDINATE GEOMETRY. at meets the latter ; then, by Prop. 1., z=h+rcos0, y=k+rsin 6; and, substituting these values in Az + By = C, we have r(A cos 0+ Bsin 0)+ Ah + Bh=C; C- Ah - Bk A cos0+ Bsin 9’ which is the required distance. Prop. XIII. 69. To find the perpendicular distance of a point (hk) from a right line (Az + By = C). Proceeding as in the last proposition, we have and therefore r= r(A cos 0 +'B sin'6) = C— Ahi Bh ox... (LY: And since 7 is perpendicular to (Az + By = C), we have, by ae 51, tan (0-2) =- 4, or cot 6 = ¥,— and therefore *r(Bcos@- Asin 0)=0. apeiciate ( Lh By Squaring and adding (1) and (2), we find eects C-Ah- Bk, — NCAP + B) ” which is the required perpendicular distance. _ * This result may be arrived at geometrically as follows: Let 4B (fig. 51) be the line (Ax + By =C), and P the point (Ak); draw A'B' through P parallel to 4B, and draw OQQ' perpendicular to both these lines; then QQ! is evidently the perpendicular distance required. Now the equations of 4B and A’B' are Az + By=C, Az + By =C’, where C'= Ah+ Bk; and we have therefore : OA OQ = OA sin QAO = V( + cot? QAO)’ Y ~ C Bs fore which, since OA = 7’ tan QAO = BR Sives | 61 OO (a) i ; at c and similarly, we find OQ = (4B) BAB)" C- C' C— Ah —- Bk ‘Hence QQ'= WATE BY * (AP + BY * Qs PLANE CO-ORDINATE GEOMETRY. 70. Cor. Hence the perpendicular distance of the line (Ax + By =C) from the origin is VA? + B’) Prop. XIV. 71. To find the equation of a right line passing through the point of intersection of the two lines (Az + By=C)...... (1) and (Az + By =C’).... (2). We may put the equation of any right line whatever in the form P(Azx+ By-C)+Q(4¢7+ By-C)=f.... (8), for we may make this equation agree with any proposed equation of the first degree, by giving proper values to the disposable quantities P, Q and R. Now, if (1), (2), and (3) intersect in the same point, they must be satisfied by the same values of 2 and y; hence, sub- stituting the values of z and y which satisfy (1) and (2) in (8), we find Ove Th. Hence (8) becomes P (Az+ By-C)+ Q(4'e+ By -C)=0.... (4), which is the required equation, and in which P and Q are arbi- trary. Indeed it is manifest immediately that (4) represents any line passing through the intersection of (1) and (2); for (4) is evidently satisfied by the values of 2 and y which satisfy (1) and (2); 7.e. the intersection of (1) and (2) is a point on (4). Various Problems respecting Right Lanes. 72. To determine the area of the triangle BPD’ (fig. 30), having given the equations of the lines BP and BP. Let the given equations be Yf = IN Oi, tc (1), I Te Ol a tenee (2), | draw PQ perpendicular to OY, and let A be the area required: then 2A = BB’. PQ, and BB = OB- OB. Hence, since OB and OB ' are the values of y got from (1) and (2) by putting z = 0, and PQ the value of z got by subtracting (2) from (1), we have _ (c= — m'-m = PLANE CO-ORDINATE GEOMETRY. 29 73. To determine the area of a triangle, having given the equations of its three sides. Let BP, BP’, BP’ (fig. 31) be the three sides of the tri- angle PP'P’, produced to meet OY in B, B’, B’; and let the given equations of BP, BP’, B’P’ be Y=Mr+e, Y=ME+C, OE Ne Mele Then, if A be the area required, we have A=BPB+BPB —BP SB’; and therefore by the ere problem a bp e I Ca AC ele -m m-m m-m 74. To inscribe a square (PP'M'M) (fig. 32) in a triangle BAA’, one side of the square being upon the side 4A’ of the triangle. Take the side 4A’ produced and the perpendicular upon it BO as co-ordinate axes; let OA =a, OA' =a’, OB=45; then the equations of AB and A’B are ctl ia ere st = GL), an ena rote I (2); and, if PM = c, the equation of PP’ is Now, since MPP'M' is a square, c= MM' = OM' - OM. Hence, since OM is the value of z got from (1) and (8), and OM’ the value got from (2) and (3), we have which gives (a' ~a).d which determines the side of the square required. im 75. This suggests the following geometrical construction. Take OQ= AA’, join QA, and draw OP parallel to QA; then P is one corner of the square. For, drawing PP’ parallel to AA’, we have PR BP.) BO MAb) BA Te Os 30 PLANE CO-ORDINATE GEOMETRY. PP b or OF eacgptlwat wer Mesa o%e oe oe ao a — eA which shews that PP’ = c. 76. B (fig. 33) is a fixed point on OY, Q is a moveable point on OX, BQP is a right angle, and BQ: QP is an invariable ratio ; to determine the locus of P. Let OB=6, QP: BQ=m:1, OM =z, and MP =y; then, since the triangles OBQ and QPM are similar, we have 0q=-4 » QM=mb, and therefore z = La mb, m m or y =m (x — mb). Hence the locus is a right line, which may be constructed thus: take OR = mOB, draw RP perpendicular to BR, and RP is the locus required. 77. To determine the equation of the right line passing through one angle of a triangle and cutting the opposite side in a given ratio, having given the equations of the sides of the tri- angle to be y cos. —‘e sin =p... . Aa odes (1), y cos ‘0 =a sth OY pie OT a eR yicos. 07.2 si, Oy= 9 Lit ete es Let the required line pass through the angle made by (2) and | (3), and cut the side (1) in the ratio m:n; and let 7’ and 7" be the distances, drawn parallel to (1) (¢.e. at an angle 0 to OX), of any point (zy) of the required line from (2) and (8). Then, by Prop. x11., we have, (putting z +7" cos 0, y +7’ sin 6 for z and y, in (2),) fix SSeS sin (0 - 6’) and similarly where, for brevity, u' = p' —- y cos 0' + x sin 6’; a ” Uu : ) Yr =————. where zu’ = p’ — y cos 9’ + x sin 0’ sin (6 — 0’) cae Me ; ‘ and we evidently have 7: 7’ :: m:n; hence mu" nu MOT: Fat) WI ak Sere ee a ee 4 sin (0-6) sin (0 - @’) a . ! PLANE CO-ORDINATE GEOMETRY. 31 m HW . y v, nr , : ! ’ or sin(0-0) (p -ysin 0'+2co0s0 = nO)? -y sin 6'+2cos 6’), which is the required equation. 78. If three lmes, drawn through the three angles of a triangle, and cutting the opposite sides in the ratios m:n, m':n’, m:n respectively, meet in the same point; then mm'm' = nn'n’ . For (4) represents one of these lines, and the two others are in like manner represented by m'u nu sin (6' — 0) cre sin (0 — 0) ene) s) alee) 06,076 (5), mu NU ain COT =O oqecia (BID Op nuny’ 4 17 te Oe where w (in the same manner as w' and w’) represents (p-y cos 04-2 sin 9). Now, if the three lines meet in the same point, we may sup- pose z and y, and therefore wu, u’, u’ to have the same values in the three equations ; and then, multiplying the three equations together, we find immediately mmm" = nnn". 79. Ifm=n and m' =n’, then this result shews that m” =n’. ‘Hence it follows that if two of the lines bisect two sides of the triangle the third will bisect the remaining side. = 80. Representing the three sides of the triangle by the same equations (1), (2), and (3), to determine the equation of the right line drawn through the angle formed by (2) and (3) and perpen- dicular to (1). __ By Prop. xtv. the equation of any line passing through the Intersection of (2) and (3) is Ply cos 9’ — zsin 0' - p’) + Q(y cos 8” - x sin 8" — p")=0...(7); and, this line being perpendicular to (1), we have P cos &+Q cos 0" | ~ Psin 0 + Qsin 6" A ‘or P cos (0 - 6’) + Q cos (0 - 0”) = 0. uw Hence (7) becomes ue u cos (0 — 0’) a cos (0 - 6") anes sitet eaten (8). 34 PLANE CO-ORDINATE GEOMETRY. 81. The equations of the other two perpendiculars are in lke manner aul! u LIPPER Tres OE Tr — TEOMA VSO @eeseeeee#e# ee 9 be cos (0'- 0") cos (0 — 8) (9) U uw cos (0"— 0) cos (0 — 6) Now if we suppose z and y, and therefore uw", to have the same values in (8) and (9), and if we multiply the two equations together, we find 7 u cos (0 — 6") cos (0 — ie which is identical with (10). Hence the same values of z and y | which satisfy (8) and (9) satisfy (10) also; ¢.e. the intersection of (8) and (9) is a point on (10). It appears therefore that the three - perpendiculars meet in the same point. 82. To determine the equation of the right line drawn | through and bisecting the angle made by (2) and (8). As before, (7) represents the line drawn through the angle formed by (2) and (8); but since it bisects the angle formed by two lines inclined at angles 0’, 6 to OX, it must make an angle gu > i with OX. We have therefore P sin 0 +Qsin 9" 0’ + 0" SPE STATES pe gee, eek REA , P cos 0+ Q cos 6" or Pn ey oe and therefore Faas Hence the equation required is a) Sar as ent Si Reb Me Biren @ bhi 83. ‘The equations of the bisectors of the other two angles are in like manner oe Ul araidntgetgny Sepak eae CN 1655 10), Che MOEN eee e's nda And reasoning exactly as in the previous case, we may — immediately shew that the three bisectors meet in the same point. It is remarkable that the equation of the line passing through PLANE CO-ORDINATE GEOMETRY. ' oo and bisecting the angle formed by the lines (y cos 9-z sin 0 = Pp) and (y cos 9’ — z sin & = p') should be y cos @- x sin 0- p=y cos f' — x sin 6 -— py’. 84. Having given a set of » points (hk), (h'k'), (hh), &e., to draw a right line in such a position that the sum of the perpen- diculars let fall upon it from the given points shall be equal to zero, (2. e. the sum of those on one side equal to the sum of those on the other). Let the equation of the required line be YeGOs Me BIN GeO ec a CL then the perpendiculars upon it from the given points are, by Prop. xtlt., p-kcos@+hsin 0, p-—k'cosO+h'sin@, p-k' cos 0+h" sin 8, Hence, if the sum of the » perpendiculars be zero, we have mp—-(k+kh +k'...) cos 8+ (h+h' +h"...) sin 0=0; or, for brevity, putting A+R +k =k, hth +h"... = nh, we have p=hk, cos 6 -h, sin 0; and, putting this value of p in (1), we have (y — k,) cos 0 - (x —h,) sin 8 = 0, which is the required equation. It is the equation of any line passing through the point (4,/,), for 6 may have any value what- ever. Hence, if we draw any right line through the point whose ; h+h'+h' + &e. k+k+k' + &e. meordinates are ———___"_ “ and » the n ) mum of the perpendiculars let fall upon it from the points hk), (WR), (WR), &e. 48 zero. | D CHAPTER IV. OF OBLIQUE CO-ORDINATES. MODIFICATIONS NECESSARY IN THE PRE- VIOUS RESULTS WHEN THE CO-ORDINATES ARE SUPPOSED TO BE OBLIQUE. POLAR CO-ORDINATES. TRANSFORMATION OF CO-ORDINATES. Of Oblique Co-ordinates. How the previous results are modified when the Co-ordinates are Oblique. 85. It is sometimes convenient to make use of axes of co-ordinates inclined to each other at some given angle different / from a right angle. In such a case, if OY, OX be the axes (fig. 84), and PM, PN respectively parallel to them, we con- sider OM and ON to be the co-ordinates of the point P. Axes — and co-ordinates of this kind are denominated oblique axes and oblique co-ordinates ; those we have been previously making use. of being called rectangular in contradistinction. 86. All that we have said, as far as the end of Art. 37, is” equally true whether the co-ordinates be rectangular or oblique, and the same may be said of Articles 43-48, 53-55, the first form of 56, 58-60, 64, and 71. In the other Articles some alteration must be made. | 87. Thus, instead of the formule in Prop. 1., we have the following (w being the angle YOX, or, as it is called, the angle of ordination), viz. fee ap Bw) an _, sin ; snw ~ LigR le”, sin w” r= (a — x) + (y'— yf + 2 (a - 2) (y'- y) cos », sin @ y-y sin(w-0) a —2’ PLANE CO-ORDINATE GEOMETRY. 35 The truth of these formule is evident from fig. 85 (which corresponds to fig. 8), in which BP =r, PQ=xz-2, PQ=y'-y, LP PQ=0,LP'QP=7-.. 88. Prop. rv. is equally true whether the co-ordinates be rectangular or oblique; only, in the latter case, instead of saying that iced of tan@=7—3 eet ep we must say, that, by Art. 87, sin 0 i fae oe A sn(w-6) 2-2 Bo 89. Hence, instead of the Cor. to Prop. 1v., we must say, that, A sin (w - 6)+ Bsin@=0; therefore A (sin w — cos w tan 0) + Btan§@ = 0; | A sin » and therefore tan § = aS ava which is the tangent of the angle the line (Az + By = C) makes with the axis of z when the co-ordinates are oblique. 90. The condition of parallelism in Prop. vit. is equally true whether the co-ordinates be rectangular or oblique; only in the latter case, instead of saying that | tan =~ <, tn 0--<, we must say, that, by Art. 87, sin @ A sin A | ra esO ae es csin a0) 4B | 91. The condition of perpendicularity in Prop. vim. must be altered as follows. By Art. 89, we have tan 0.(4 cos w - B) = A sin », tan @.(A' cos w- B’) = A’ sino; nd hence, since tan 9. tan 0’ =-—1, we have (A cos w — B) (A’' cos w — B') + AA’ sin’ ow; or A’ (A - B cos 0) + B (B- A cos ») = 0, hich is the required condition instead of (2), Prop. vit. a D2 36 PLANE CO-ORDINATE GEOMETRY. Hence it follows that -(B- A cos w)x+(A- Boos w)y=0 is the equation of a line perpendicular to the line (Az+ By =C),) the co-ordinates being oblique. | 92. Prop. x. must be altered as follows. By Art. 89, we have A’ sin w A sin A’ cosw- Bo Acosw-B A’ sin w A sin w | tan p = 1 + cos one A Cos aieee c (A B'— A’'B) sin w F ” AA’ + BB -(A'B+ BA) cos 0 | From this expression, by putting ¢ = 0, or Ne we obtain the conditions necessary in order that two right lines may be) parallel or perpendicular to each other. . 93. Lastly, Props. x1. and x11. must be altered as followsét By Art. 87, we must substitute in the equation Az + By = the values ; Hi sin (w — @) Ch a7 y=kr+r — e 2 sin w SIN w | mn and then we have oe r {Asin (w- 6)+ Bsin 6} = (C- Ah- Bh) sinw .... (1), which gives +. | If r be perpendicular to (Az + By =C), 0 - 5 must be the angle which the latter line makes with the axis of x; and there- fore, by Art. 89, we have A sin w - Toor 7 (0-F)=-co8........@ If for a moment we put B- A cosw=A, Asinw =u, (1 and (2) x r become r(u cos 9+ A sin 6) = (C- Ah - Bh) sin o, r (un sin # — Xd cos 0) = 0; Ce ~t PLANE CO-ORDINATE GEOMETRY. rence, squaring and adding, we find r (rN? + pw’) =(C- Ah -— BRY sin? o ; | ore pu AGA - Bh) sino ud therefore 7 a Lala 0 eee ES By which is the expression for the perpendicular distance of (hk) tom (Az + By = C), the co-ordinates being oblique. Of Polar Co-ordinates. 94. Besides the method of rectangular and oblique co-ordi- lates just explained, there is another method often employed, aamely, that of polar co-ordinates, which consists in representing che position of a point in the following manner. Let P (fig. 36) be the point whose position we wish to repre- sent ; choose any line OA and any point O upon it, join OP, wsume OP =r and 4 POA = @; then the position of the point P is given when the values of the quantities r and @ are given. These quantities are called the polar co-ordinates of P; O is valled the pole, r the radius vector, and OA the prime rie 9 is sometimes called the vectorial angle, though it is not usually distinguished by any particular name. 95. ‘The sign - is applied to polar co-ordinates on exactly the same principles as those already explained in the case of rectangular co-ordinates. Thus, if + represent any distance measured from O towards P,—r represents an equal distance measured in the opposite direction from O towards P’; and if 0 represent any angle measured from towards P, — 6 will repre- sent an equal angle measured in the opposite direction from A towards Q. A few examples will make this clearer. 96. Let @ be any distance measured from O towards B, 0 being the angle which OP makes with OA (see Art. 39); then Ue r =a represents P in fig. 37, = Tee Meee sees ES OS 0 Tielke hep. en. 1s. 89, A= 7 POS PAs Re cer el wh eal fig. 40, 6) 38 PLANE CO-ORDINATE GEOMETRY. It is important to observe that the direction in which r is measured depends, not only upon its sign, but also on the value of @; thus when @ = = and 7 = — a, r must be measured from O towards P (fig. 39); and when 6 = a and r = a, r must be measured in exactly the same direction. Again, when @ = 0 and y =a, and when 6 = 7 and r = — a, r must be measured in bot cases from O towards A. PRaPr OV? 97. To determine the distance between two points whose polar co-ordinates are (7, 0), (1’, 6’). Let P, P' (fig. 42) be the two points, POA = 0, POA = 0 OP =r, OP'=7'; then in the triangle POP’, we have PP* = OP’ + OP” -20P.0P' cos P’OP, or PP =r +7? — 2rr' cos (0 - 6), which is the required distance. Prov. XVI. 98. ‘To determine the polar equation of a right line. In the same manner that curves and lines are represented by equations between z and y, they may also be by equations between rand @, which are called polar equations. Thus, let BP (fig. 43) be any right line, P any point of it, OP (= r) and POA (= 6), the polar co-ordinates of P; and let OB=a PBA=f[; then OP: OB::sin PBA: sin BPO, Pe sy 6: i a sin (GB — 6)’ which is a relation between the polar co-ordinates of any poin of the right line, and is therefore the polar equation required. | 99. Cor. This equation may be put in the form, Ar cos 0+ Brsin 0=0, which is therefore the general form of the polar equation of right line. PLANE CO-ORDINATE GEOMETRY. 39 Transformation of Co-ordinates. 100. It is often necessary to change, or, as it is called, to transform the co-ordinates which represent the position of any point from oneset to another; for example, supposing the position of the point expressed by rectangular co-ordinates, to express it by oblique or polar co-ordinates. We now proceed to shew how transformation of this kind may be effected in various cases. Prop. XVII. 101. To transform polar co-ordinates into rectangular, or vice versd. Let P (fig. 48 b¢s.) be any point, and OP (=r), POA (= 86) its polar co-ordinates; take OAX as axis of z, and OY drawn at right angles to OA as axis of y; draw PM perpendicular to OX, and then OM (=z), MP (=y) will be the rectangular co-ordinates of P. Hence we have | Pe rACOseG? Lite arsine 0 tn. ce CL), which give z and y in terms of r and @; also r=V(2'+y’), tan 0= é ban Se ae Ae} which give r and 6 in terms of z andy. And thus, from the polar we may obtain the rectangular co-ordinates of a point, or vice versa. Hence, by making the substitutions (1) in any equation between z and y, we change it into a polar equation; and by making the substitutions (2) in an equation between r and 0, we change it into a rectangular equation. For example, by putting x =r cos 6, y=r sin 8, in the equa- tion Ax+ By =C, it becomes Ar cos 0+ Br sn 0= C; which agrees with the result of Art. 99. Prop. X VIII. 102. To turn rectangular axes round their origin through any required angle (a). _ ~Let z and y be the rectangular co-ordinates of any point P (fig. 44), and r and @ its polar co-ordinates, as in the last article; then z=rcos@=r cos(@-a+a), =r cos(9-a).cosa—rsin(9—a).sina, and y=rsin@=r sin(@-a).cosa+rcos(@—a).sina. 40 PLANE CO-ORDINATE GEOMETRY. Now rcos(@-a) and rsin(O—a) are evidently the co- ordinates of the point referred to the rectangular axes OX’, OY’, £X'OX being equal to a; call these co-ordinates “4 and y'; then z= 2 cosa—y' sin a, | y=y' cosa+z' sina, | Hence, if we have any equation between z and y, and we make in it the substitutions (1), it becomes an equation between zx andy’. This process, therefore, turns the axes through an angle a; ¢.e. it introduces co-ordinates referred to the axes OX', OY", in place of those referred to OX, OY. | 103. For example, let the equation between a and y be gel Ra blipal webby Vey 0 3 Bis, (PNP and let a= 45°; then the equations (1) become e=— @'-y), y= (@+y); hence (2) bécomes (v-y'l-@+y'f = or 2EY bait =) Eten: (3). - Thus theequation (2) referred to OX, OY, is equivalent to | bo g aS) the equation‘(8) referred to OX', OY’, £ X'OX being 45°. 104. In making the substitutions (1) we may omit the dashes for the same reasons as those explained in Art. 58, and_ then we may enunciate the result of this Article in the following manner. 0 turn the axes of co-ordinates through any angle (a), we have only to put 2 cosa—y sina in place of z, and y cosa+z sina in place of y; remembering that z and y, after these substitutions, are referred to OX’ and OY’. Prop. XIX. 105. ‘To transfer the origin to a point (44) without altering the direction of the axes. Let O'X', QO'Y’ be the new axes (fig. 45) which are respec- tively parallel to OX and OY, and draw PM'M parallel to OY; then OM =0Q+ O0'M', MP=Q0'+M'P;; or, since O' is (hk), thie, yakry'; PLANE CO-ORDINATE GEOMETRY. 4] which substitutions, made in an equation between z and y, change it into an equation between z' and /. Hence, suppressing the dashes, as in the last Article, it appears that we transfer the origin to the point (h/), without altering the directions of the axes, by putting 4+ in place of « and & + y in place of y. 106. Cor. 1. Hence, to transfer the origin to a point (h/), and to turn the axes through an angle a, we have only to put h+2cosa—ysina in place of z, and 4+ycosa+z sina in place of y. 107. Cor. 2. In the case of polar co-ordinates it appears, in exactly the same way, that to turn the prime radius through any angle (a), we have only to put 6+ for 0. Prop. XX. 108. ‘To reverse the positive direction of the axis of x, or of the axis of y. __ If we suppose OX’ (fig. 2) to become the positive axis of x instead of OX, every distance measured along X’X will become the negative of what it was before. Hence it is evident, that if we write — 2 for z in any equation, it amounts to reversing the positive direction of the axis of z. And in the same way, if we write —y for y, we reyerse the positive direction of the axis of y. 109. Cor. In the same manner, by writing — 6 for 0, we teverse the positive direction of the vectorial angle. EROE AL _ 110. ‘To transform the co-ordinates from one set of oblique axes to another having the same origin. Let OX, OY, and OX’, OY" be the two sets of axes (fig. 46), OM, MP, and OM’, M'P, or z, y, and z’, y' the co-ordinates of any point P referred to them; draw M'Q parallel to OX and M'R parallel to OY. Then we have OM= OR+ M'Q - om’, BY, yp, 29" sin (xy) "sin (ay) 42 PLANE CO-ORDINATE GEOMETRY. [We mean by sin (zy) the sine of the angle which the axis of x makes with the axis of y, and by sin (2'y) the sine of the angle which the axis of z’ makes with the axis of y; and so with the rest.] Similarly, MP = RM'+QP Mw sin ee) ip sin (y'x) ; sin (xy) sin (zy) Hence, if we assume Z(zy)=0, ZL(¢'z)=a, L(y'x)=8 we have , sin (w - a) ay sin (w _ 2) A by ==) 4b > ° 2 Ss1n w SIn w sin a sin Ye ei fee LY sin w sin w which are the formule necessary to effect the required trans- formation. 111. Cor. If we wish at the same time to transfer the origin to a point (hk), we have only to add h and & to these expressions for z and y. Prop. XXII. 4 112. In the case of polar co-ordinates, to transfer the pole to any point (24), and turn the prime radius through an " angle a. | It is easy to see that we effect this transformation by mean of the formule r cos 0=h-+7' cos (0 + a), r sin @=4+7' sin (@' +a), which will enable us to change an equation between r and @ into one between 7’ and 6’; it being evident that r’ and @' are the polar co-ordinates referred to the pole (hk), and to a prime radius making an angle a with the original prime radius. The use of these various ways of transforming co-ordinates will appear as we go on; we shall often have occasion to refer ta Props. XVIJ., XVIII., XIx., and xx. PLANE CO-ORDINATE GEOMETRY. 43 Various Problems respecting Lines referred to Oblique Co-ordinates. 113. If O (fig. 47) be the middle point of the side YY’ of any triangle YX Y’, and if we draw Y'’P, YP’ through any point M of OX; to shew that PP’ is parallel to YY’, and that OX is divided harmonically at the points M and N, (v.e. the reciprocals of OM, ON, and OX are in arithmetical progression). Take OX and OY as co-ordinate axes, and let OY = OY'=8, OM=a, OX =a’; then the equations of Y’P and YX are respectively a Be gg ihe a —b ane Aime ies Iel4 @\ ytd ___ If we suppose z and y to have the same values in these equa- tions, (zy) is the point P; therefore, adding the two equations, which give P a (7 ~ =| LG RE AW maces Aarne (1), | a a the value of z we determine is the abscissa of P. Now, in exactly the same way we find the abscissa of P’ by adding the equations of YP’ and Y'X, viz. i a aH Sas I] a b x y sth ed Sige seen pA AP a ees b : which give o + a eee PAE g fe ghee) Aiawee oil * : (1) and (2) shew that the points P and P” have the same abscissa, and therefore that PP’ is parallel to YY’. Q.E.D. Again, (1) gives dees): ay: Si L ] rag, >) a a , | ee ; : 1 1 which shews that — is an arithmetic mean between ; and a and x U . . . therefore that a, z, @ are in harmonical progression. Q.E.D. 44 PLANE CO-ORDINATE GEOMETRY. 114. If OX, OY (fig. 48) be any two right lines intersecting a set of diverging right lines PB, PB’, PB", &c., and if OX be divided harmonically by them; then will OY also be divided harmonically by them; ¢.e. if OA, OA’, OA", &c. form a har- monical progression, so also will OB, OB’, OB", &c. Let OA = a, OA' = a,, OA" =a,, &., OB = b, OB' = b, &e.; and let(hk) be the point P. Then the equations of AB, .A'B',A’B’, &c. are ee Gti a if b ‘ x Y = — = ] a, + b, OC ace, and since these all meet in P, we have Pe ys ees at a i b i h ~b k = ha a, 6, Odea oat) By subtracting any one of these equations from that which follows it, we find . 1 it 1 1 ; (g-- a) Bt (Ge - ge) Be Ore. Pie: Now, since a, a,, a,, &c. are in harmonical progression, 1 1 — — — is an invariable wero! for all values of 2; hence, a n-1 by (1), the same is true of = 5, and therefore 5, 6,, b,, &c. n-l n are also in harmonical progression. aq. E. D. Hence, if OX be cut harmonically, every line drawn from O is also cut harmonically by the diverging lines. : 115. OX, OY, and BA (fig. 49) are three given lines, and A'B' is any other line cutting them in 4’, P, B’; if OA’ + OB be always equal to O.A + OB, then will B’P: B’A’ be an invari-- able ratio. PLANE CO-ORDINATE GEOMETRY. 45 Take OA and OB as axes, and let OA =a, OB=b, OA'=d, OB = b'; then the equations of 4B and A'B’ are MT coata cttictetehs ake coe rete A aie (1) A + f ZA Nb et brat eat rary eA ea) a Supposing z and y to have the same values in both these equations, which amounts to supposing (ay) to be the point P, oe find, eliminating y, a(2- 5) 5-85 a a 6 Bb ba-ba_ b(a+b6-8)-Jda CECT TET aE (6 — 5') (a+) ite hE capa x a Hence es aes / a @+b% Now 2: a':: B'P: B'A'; hence B'P: B'A' is an invariable ratio. 116. If ABC be a triangle having its angles .A and B always upon two fixed lines OX, OY, and if the three lines which form the sides of the triangle always pass through three fixed points which are situated in the same right line; then the angle C’ will always be found upon a fixed line passing through 0. Let the equations of AB, AC, and BC, referred to OX and OY as axes, be alt ciy GEIR RAL x | AGRO Pa Ca en (1), at iay POD er apart le mage (2), RO fl ed Wepre RRA Sear Le (3). Observing that, since (1) and (2) meet in OX, they must give the same value of z when y= 0; and for a similar reason (1) and (3) must give the same value of y when z = 0. Let (hk), (A'h') and (hk) be the three points through which (1), (2) and (3) always pass respectively, and let the equation of the line in which these points are situated be PLS IULE Sar Lay obo « sire, 9a Saco ate (4). 46 PLANE CO-ORDINATE GEOMETRY. Then, since hf is a point on (1) and (4), we have, subtracting (4) from (1), (a-~-m)h+(B—-n)k=0; and similarly, since (A'k') and (h’k’) are points on (2) and (4) and on (8) and (4) respectively, we find (a-m)h' +(B' —n) kh =0, (a -m)h'+(B-n)k = 0. These three equations shew that, whatever be the positions of the lines (1), (2), (3), the quantities (a - m), (6 — »), (a’ — m) and ((3'—) are to each other in invariable ratios depending upon the quantities Ak, hk’, h’k’. Hence it follows that (a — m) - (a - m) : (8 —n) — (B' -— x) is an invariable ratio, A: B suppose ; and therefore we have a a A Bea abe tee (5). Now, supposing (ay) to be the point of intersection of the lines (2) and (3), we have, subtracting (3) from (2), (a-a')2-(B- By =0, which, by (5), becomes AL. BY =A) 0). cha ky See (6). @ Hence the point of intersection of (2) and (8), ¢.e. the point — C, is always situated on a fixed line passing through the origin, | whose equation is (6). . 117. Hence the truth of the following theorem is immedi- 3 ately evident, viz. If ABC, A'B'C’ be two triangles such that the point of intersection of AB and A’'B’, of AC and A’C’, and of BC and BC", lie in the same right line; then the three lines 4.4’, BB’, CC’ will meet in the same point. CHAPTER V. GENERAL EQUATION OF THE CIRCLE. DIAMETER, TANGENT, AND NORMAL OF A CIRCLE. VARIOUS PROBLEMS RESPECTING CIRCLES AND RIGHT LINES. Prop. XXIII. 118. To find the equation of a circle whose centre and radius are given. Let (Ak) be the centre, a the radius, and (xy) any point of the circumference ; then the distance of (zy) from (hk) must be #, and therefore, by Art. 38, we have (Zi Aa fm BY St Ohi vciviny ould os sG1p, which, being a relation between the co-ordinates of any point of the circle, is the equation required. 119. Cor. 1. If h=0, 4 =0, @.e. if the centre be chosen as origin, (1) becomes e+ = a’. 120. Cor. 2. Ifh=a,k=0, 7@.e. if a diameter be chosen as axis of x, and one of its extremities as origin, (1) becomes yo = Loan? a. 121. Cor. 3. The equation (1) may represent any circle whatever, and therefore it is the general form of the equation of acircle. If we expand each term, it becomes a +y’ —2he-2hy+h+kh-a@=0; and hence it appears that the general equation of a circle is of the form a+y’+ Ax+ By + C20.......... (2), A, B, C being any arbitrary constants. The equation Ax’ + Ay’? + Be+Cy+D=0........ (3) may be immediately reduced to the form (2) by dividing it by 48 PLANE CO-ORDINATE GEOMETRY. the coefficient of 2” and y’. (8) is the most general form in which the equation of a circle can occur, the co-ordinates bein rectangular. Prop. XXIV. 122. Conversely. ‘To determine the locus represented b the equation a+y+ Ax+ By + C=0. : 2 2 Completing the two squares by adding =a and = to both sides of this equation, we have Noiich hy ab wigirr ar: 2+—)+ly+—] =—4+—-C4 2 2 4 4 Now the first member of this equation is the square of the distances of the point (zy) from the point (- 2 , 7 >) : hence the locus is a circle whose centre and radius are 2 2 , Ga - 5) ana Vie+s-¢): ; 2 2 Bs 4 4 123. For example, let the equation be x+y’ — 2cx + 6cy + 9c" = 0. Completing the squares, we have (a-cl+(yt+ 8ey =e’, which shews that the centre of the circle is (¢, - 3c), and the radius ec. Again, let the equation be z+ y’ — 2cx -— By + 2c. ee eighteen F4 Completing the squares, we have (c-e)+(y-cy =0. Here the centre is (cc), but the radius is zero; therefore the locus is the single point (cc). Lastly, let the equation be xv +y’- 2cx + 4cy + 8c' = 0. Completing the squares, we have (a - cy +(y + 2cf =- 8c" Here the radius is impossible, and therefore the equation doe not represent any locus. : Lan _ i > ‘PLANE CO-ORDINATE GEOMETRY. 49 Prop. XXV. 124. To find the equation of a circle referred to oblique co-ordinates. Expressing the distance of the point (zy) from the point (hh), by Art. 87, we have (a@-h) + (y- ky + 2(@-h) (y-k) COS w = a’, which is the equation required. -125. Cor. If the centre be the origin, the equation is z+ y’ + Icy cos w = a’. | Prop. XXVI. 126. To find the equation of a circle referred to polar co- ordinates. Let r and @ be the co-ordinates of any point of the circum- erence, and 4 and (3 those of the centre: then, by Art. 97, we faye r + 6’ — 2rb cos (0 - B) = a’, vhich is the equation required. 127. Cor. 1. If 6=0, ze. if the centre be origin, the equa- ion becomes pai Cor. 2. If b=a and B=0, é.e. if a diameter be the prime adius and one extremity of it the pole, the equation becomes r = 2b cos 0. Diameters, Tangents, and Normals of a Circle. Prov. XX VII. 128. To determine the length (7) of a right line drawn, at n angle @ to the axis of z, from a point (hk) to the circum- 2rence of a circle (z’ + y’.= a’), (see Art. 68). Proceeding exactly as in the Article just referred to, the quation of the circle, when h +r cos @ and k+/,r sin 6 are put x z and y, becomes (h+rcos0)+(k+7 sin 0Y = a’, r+2(heos0+hksin 0)r+h+h-a=0....(1), ‘hich equation determines the required distance ( Yr). 50 PLANE CO-ORDINATE GEOMETRY. Cor. 1. Since this is a quadratic equation, it follows that the right line must, in general, meet the circle in two points, whose distances from (hk) are the two roots of (1). 129. Cor. 2. Let P (fig. 50) be the point (Aé), and Q, Q the two points where the right line meets the circle; then PG and PQ’ are the two roots of (1), and therefore PQ+PQ =-2(heos0+ksinO)...... (2), | PQ. PQ aK ais oa - ee (3). 130. Cor. 3. From (8) it follows that, if through any poin{ P asecant be drawn meeting the circle in Q and Q,, the rect angle under the segments PQ and PQ’ is invariable in whateve} direction the secant be drawn. (Huchd, Book 111. Props. ai and 36.) 131. Cor. 4. If R be the middle point of the chord ad we have 2PR= PQ+ PQ, and therefore, by (2), PR==(h'cos 0+ hein 0)... 2.254 (4). 132. Cor. 5. Hence, if P and R coincide, in which casi PR = 0, we have Acos,0 +-4sin. 6 = 0 ee (5).8 ; which is therefore the condition necessary in order that (hé) may be the middle point of the chord which makes an angle 6 wil the axis of z. i This appears also from the fact that, if (5) hold, the equation (1) gives r=+V(e-B-B), : which shews that one value of 7 is the negative of the other, a f this cannot be unless P be midway between Q and Q’. q Prop. XXVIII. 133. If a chord of a circle be supposed to move ee itself,* to determine the locus of its middle point. vi r, i Let @ be the invariable angle which the chord makes witl the axis of z, and let (zy) be its middle point; then, by th 5th Cor. of the preceding proposition, we have g xzcos 9+ ysin 6 = 0, * ze. always making the same angle with the axis of z. PLANE CO-ORDINATE GEOMETRY. 51 which, being a relation between the co-ordinates of the middle point of the chord in any of its positions, is the equation of the required locus. Hence the locus is a right line, evidently passing through the origin, and making an angle whose tangent is — cot 8 (: e. an angle 6 + ; with the axis of z. 134. Cor. The locus of the middle points of a system of parallel chords is called the diameter of those chords; hence, by what we have just proved, the diameter of any system of parallel shords in a circle, is a right line passing through the centre, and verpendicular to the enn ds. Prop. XXIX. 135. To find the angle which the line touching a circle at a yiven point makes with the axis of z. If a right line cutting a circle be supposed to move parallel 0 itself, until the two points of section coincide and become one yoint, it is then said to fowch the circle at that point, or to be the angent of the circle at that point. If we suppose Q and Q' (in Art. 129) to coincide, we have PQ = PQ’, and roe by equation (2) in that article, we have | Q=-(hcos 0+ zk sin 8), vhich, since the secant is now become a tangent at the point Q, 3 the distance of any point (P) on a tangent from the point of ontact (Q), 6 being the angle which the tangent makes with the xis of z Now suppose P to coincide with Q, in which case ?Q = 0, and (4k) becomes the point of contact ; then we have hcoos#+ksn@=0, or tan 0=-*, thich is therefore the tangent of the angle which the line touch- ag the circle at the point (h4) makes with the axis of z. 136. Cor. 1. Hence the equation of the tangent at the oint (hk) is, by Art. 61, h(a -h)+k(y-k) = 0, thich, since h’ + h” = a’, (hk) being a point of the circle, becomes RGarikiidi die, acts binkt ohiemdnel > (1), 52 PLANE CO-ORDINATE GEOMETRY. 137. Cor. 2. To determine the condition necessary in order, that the line (az + By = y) may touch the circle (2 + y’= a). Let (hk) be the point of contact, then the equation az + By = must be identical with (1),* and therefore we have Aj aia ee Oe Qa Y a Y d and hence z aig = ie ul ig 7 Ee , y a a or (a” be 2”) a = Ves which is the condition required. 4 Cor. 3. Hence, if the line z sin 6- y cos 0=c be a tangent to the circle, we have } c= a’ (cos’ 9+ sin’ 0™)=a@"*, or c=+24; and therefore the equation becomes zsin@-—ycos0=+4, ; which is the equation of the tangent in terms only of the angle it makes with the axis of z. This is a useful form of the equa- tion in problems where there is no occasion to bring the point of contact into consideration. The double sign indicates that two. different tangents may be drawn making the same angle with the axis of z, which is manifestly the case. 2 Prop. XXX. 138. To find the equations of the normal at any point (hh) of a circle. } The right line drawn through the point of contact at right angles to he tangent is called the normal. ? The equation of the line passing through the point (AA) at right angles to the tangent (hx + ky =a’)is, by Art. 63, and Art. 60, : -k(w-h)+h(y—-k) = 9, : or ka —hy = 0, i which is therefore the equation required. It shews that the * Tn order that two equations Ax + By =C and d'x + By = C’, may b identical, it is mot necessary that A, B, C shall be respectively egual t A’, B', C, but only that they shall be respectively proportional to each othe PLANE CO-ORDINATE GEOMETRY. 53 aormal always passes through the centre, and therefore it follows shat the tangent at any point is at right angles to the radius lrawn to that point. (Huchd, Book m1. Prop. 16.) Prop. XXXI. _ 139. ‘Two tangents may be drawn to a circle from any point Ak) without it ; and the equation of the right line joining the Memes contact isi A> angie gil ee) Sober (ny, For, let a right line drawn through (A&) touch the circle at he point (/’x'); then the equation of this line, by Art. 136, is re+ k'y =a’, and this equation must be satisfied when h and & re put for z and y, since (AX) is a point on the line; therefore ve have, observing that (/’/') is a point on the circle, tL lata te tal Cy, [ier oss aaah Me lake (3). f in (3) we substitute for #’ its value given by (2), we evidently md a quadratic which gives two values of #'; and then (2) gives wo corresponding values of k’. Hence there are two points of ontact, and therefore two tangents may be drawn from (hk) to he circle. _ Now let (h’h') be the other point of contact thus determined ‘om (2) and (3); then we have, by (2), OLE KELP DHE OIV LE BURNS. (4). lence it appears that (h’2') and (h’h’) are points on the right ne represented by the equation ) NE ae Nat OS A Ree (53 m (2) and (4) shew that (5) is satisfied when either h' and &’ or ‘and ’ are substituted for x and y. _ Hence it follows that,(5) is the equation of the right line iming the points of contact. @. E.D. Prop. XXXII. 140. Supposing a chord of a circle to be drawn through a xed point, and to turn round it; to find the locus of the inter. ction of the two tangents drawn at its extremities. _ Let (zy) be the point of intersection of the two tangents, and f) the fixed point through which the line joining the points of 54 PLANE CO-ORDINATE GEOMETRY. contact always passes; then, by the result (equation 5) of the last proposition (observing that (hé) in equation (5) is the pom of intersection of two tangents, and (zy) any pot on the ling joining their points of contact), we have, putting (AA) for (xy and (zy) for (hh), ha + hy = a’. | This, being a relation between the co-ordinates of the poin of intersection of the two tangents drawn at the extremities 0: any chord passing through (h&), is the equation of the locus required. The locus is therefore a right line, evidently perpen- dicular to the line - ‘ = 0, z.e. to the line joining the centre and the point (hA).* 141. It is remarkable that the equation (hz + ky = a’) repre: sents three very different lines, namely, the tangent at (AA), the line joining the points of contact of the two tangents drawn through (44), and the locus of the intersection of the two tam gents drawn at the extremities of any chord drawn through (hh) We aot observe that in the first case ae must bea point: on Various problems respecting circles and right lines. 142. To find the locus of the middle point of a chord in ; circle which always passes through a given point and turn: round it. ; * This result may be proved geometrically, thus : Let C (fig. 52) be the centre of the circle, B the fixed point through whiel the chords pass, Z"B7' any one of the chords, and P the intersection of the tangents at its extremities; produce CB, draw PA perpendicular to it, ant let PC meet TT’ in Q. Then, since CTP and CQT are right angles, we; have CQ. CP=CT"; and, since Q and 4 are right angles, we have ve CQ.CP=CB.CA; hence ~ CT CA= CB Therefore P is always found on a right line drawn through a fixed poif A at right angles to CA. 2 = a constant quantity. a PLANE CO-ORDINATE GEOMETRY. dD iP Let (zy) be the middle point of the chord, and @ the angle it makes with the axis of z; then, by Art. 182, we have POCONO USI UO abe cde sietere tate | Liye Also, if (hk) be the given point through which the chord always passes, (zy) must be a point on the line drawn through (AA) and making an angle @ with the axis of x; therefore y—k ae ate al r- h tan 0 y 3 y ( )s y —-ky+x-hx=0, which, by Art. 122, represents a circle whose centre is & e and (radius)’ feos . Hence, if Cbe the centre of the original circle, and P the point (A&), we have only to describe a circle on CP as diameter, and it will be the locus required. _ 148. Having given the equations of two circles, to find the equation of the right lime passing through their points of intersection. Let the equations of the two circles be PX AL + BY HO! oe oe tac ebeh C1) TAD PIA De BAEC? svi esa ace severe (2), then, subtracting these equations, we find CA RANGE UD SED yi CLA coe ee ets (3), which is the equation of the line required ; for we have obtained ‘8) from (1) and (2) on the supposition that z and y have the same values in (1) and (2), 2. e. that (zy) is either of the points of intersection of the two circles; therefore (3) is satisfied by the co-ordinates of either point, and therefore the right line repre- sented by (3) is the line drawn through the two points of inter- section of (1) and (2). See Art. 139. 144. To shew thatif a circle of variable radius be described, ‘touching a given line at a given point, and cutting a given sircle, the line drawn through the points of intersection always passes through a given point. Take the given line as axis of z, and the point where the 56 PLANE CO-ORDINATE GEOMETRY. circle touches it as origin; then, if r be the variable radius of that circle, its equation is Tet O0Y wen. Oilers seeks See Also, let the equation of the fixed circle be w+ y’ - 2he - Iky =c....... . (2); then, by the previous problem, the equation of the They joing the points of intersection of (1) and (2) is he +(k—r)y + = = 0. Now this line manifestly cuts the axis of x at a distance — — Cc) from the origin, whatever be the value of r ; therefore the line joming the points of intersection always passes through th e | c ive t{-—, 0}. given poin ( a, 145. To find the locus of a point (P) the sum of the squares of whose distances from a given set of points (A, A’, A’, &c.) i 1s given. ; Let (zy) be the point P, and (hh), (h'k') &c. the points -A, A, , A", &c.; then we have (x —hy +(y-ky + (@-hfP+(y-kYP + (w-h’fP t+ (y-kY 4 &e. = a given quantity, C suppose. “he ' Therefore, if 2 be the number of the given points, and if we put, ; ae +.&¢. =H, k+tk +k' + &.=K, t —(h’ +h? + h*...)—- C+ kh + kh...) = LZ, $ we find nx’ + ny’ —- 2Hxe -2Ky=L; hence the locus required is a circle. 146. ‘To find the locus of the vertex of a triangle whose base and ratio of sides are given. Take the given base as axis of x and its middle point as_ origin, let 2a be its length, and m:n the ratio of the sides; then, if (zy) be the vertex, the lengths of the sides are V{(x - a) + y*} and V{(~ + ay + y’}; and we MSs have i PGA, Rp tee (Br ay+ ys. (2+ay+y n me ? or (x + y*) (® — m*) — 2a(n’ + m*) a+ a(n? — m’), PLANE CO-ORDINATE GEOMETRY. O7 This is the equation of a circle whose centre is on the axis of | : n+ m* fe: iy RK r at a distance a. a spee from the origin, and whose radius is m 22m “nm 147. A right line is drawn in any direction from a given soint to the circumference of a given circle; to find the locus of he point which divides it in a given ratio. Take the given point as origin, and let the equation of the Meee be (zp AY + (Y— hPa ew res enna) hen if (2'y’) be the co-ordinates of the point which divides any ine drawn from the origin to this circle in a given ratio, it is svyident that the co-ordinates of a point of the circle will be mz’, ny’, m being a given ratio. Therefore these values put for # and yj must satisfy (1); and consequently we have (ma — hf + (my' — ky =, or, neglecting the dashes, ‘3 h\’ . kV @ m I im) ~ mi? ce ahh ah toad which represents a circle whose centre is | — — } and radius — . m m m [his problem may be easily solved by using the polar equation of the circle. 148. To find the locus of the vertex of a triangle whose base and vertical angle are given. 149. To find the locus of the centre of the inscribed circle, imder the same circumstances. 150. To find the locus of the intersection of the perpen- liculars let fall from the extremities of the base upon the opposite sides, under the same circumstances. 151. To find the locus of the intersection of the lines drawn from the extremities of the base bisecting the opposite sides, ander the same circumstances. If 6 and 6! be the angles which the sides make with the base taken as axis of x and its middle poit as origin), it 1s easy to 2 . 58 PLANE CO-ORDINATE GEOMETRY. shew that the equations of the two bisectors are (2 cot 0’ —cot #)y=7+4a...(1), (2 cot @-cot O')y=2- negative quantities ; we shal Fl put the equation aie in the form 2 > pep ean aed toe 0; 9 ~ e-—1 e—1 and we shall assume tray yf a fay PLANE CO-ORDINATE GEOMETRY. G5 md removing the origin to the point (- a, 0), which is done by writing z — a for z, we have 2 2 wy —_ = |] elas) a6) GB, 0 88 » ete we? 6 . = - 5 (3) This is the equation of the locus referred to a new origin O' fig. 55) where OO' = - a, the value of a being ® being a’ (e* — 1). We may shew, exactly as in the first case, that the curve is serfectly symmetrical with respect to the axes of x andy. Also, ‘ince from (8) we have Mo y=) (5-1), t is evident that, supposing WZ to move from O' in the positive lirection, MP is impossible while MZ is between O! and O, is “ero when M is at O, and continually increases when M is yeyond O. Hence the form of the curve is that represented in fig. 55. a : ; and that of 169. Thus it appears that the equation (1) represents three urves of very different form, according as z is less than, equal to, x greater than unity. In the first case the curve is called an Uipse, in the second a parabola, and in the third a hyperbola. Che origin of these names was as follows. 170. Let OHH'M be a rectangle (fig. 56) constructed ipon the abscissa 4M, and having its erect side OH equal to the coefficient of z in equation (1), namely, 2 (1 +e) m; which quan- ity has on this account acquired the name of the datus rectum. chen 2 (1 +) mz is equal to this rectangle, and therefore, by Mhwehave = yp? _ OHH'M = (¢ - 1) 2°. Hence it is evident that, according as e is less than, equal to, r greater than unity, the square of the ordinate falls short of, is qual to, or exceeds the rectangle under the abscissa and latus ectum. It was in reference to this defect, equality, or excess, aat the names ellipse, parabola, hyperbola, were given to the aree curves we have been considering. 171. The point S (Art. 165) is called the focus, and the line 1K the directriz ; Swas formerly termed punctwm comparationis. E ! 66 PLANE CO-ORDINATE GEOMETRY. The ratio e is called the eccentricity, the reason of which name will appear as we goon. Hence the eccentricity is less tha unity in the ellipse, greater than unity in the hyperbola, and) equal to unity in the parabola. . If we take O'S’ equal to O'S (figs. 54 and 55) and O'E" = O' it is evident, from the perfect symmetry of the curve with respect to the axis of y, that the point S’, and the right lne (E'K") drawn through £’ at right angles to O'E’ have exactly the same properties, with reference to the curve, as the point & and the line H#; and therefore the distance of any point of the curve from S" is to its perpendicular distance from £’ A’ in the ratioe: 1. There are therefore two foci and two corresponding directrices in the ellipse and hyperbola. 4 172. The point O' is called the centre, on account of the symmetry of the curve with respect to the two axes which inter= sect in O’. We shall shew that O' bisects every line drawn through it in both directions to the curve. It is usual to denote the centre by the letter C, the point O by the letter A, and the other point where the curve crosses the axis of z by A’. The line .A' (which = 2a) is called the azis, and the points A and A, the vertices of the curve. Cis the middle point of 4A’. E 173. In the ellipse OO' is drawn in the positive directio A, and in the hyperbola in the negative, as we have shewn above, The parabola has no centre, since OO’ is infinite; and it meets the axis of z only at O; therefore its axis is infinite, and it has but one vertex. 174. In the ellipse the pomts where the curve crosses t e axis of y are usually denoted by the letters B and B’; the line 2 BB (which = 26) is bisected by the centre. Since b = 1 a 2 (see Art. 166) and eis<1,is>a@; on this account 4A’ (or 2a), is called the major axis of the ellipse, and BB' (or 2d) the minor aXIS. 175. In the hyperbola the curve does not cross the axis Of y; on this account 4A’ is called the possible axis of the hyper bola, and the axis of y is called the impossible axis. Sine PLANE CO-ORDINATE GEOMETRY. 67 3 = @ -1in the hyperbola (Art. 168), and e may be of any magnitude greater than unity, it is evident that } may be either greater or less than @ in any proportion. 176. Figs. 57, 58, 59 represent the three curves marked with their proper letters; to which notation we shall alw ays adhere. It is important to bear in mind the following relations which are assumed in Arts. 166 and 168; viz. m AS=m AC= l—e = «ain the ellipse, = — @ in the hyperbola ; —, = 1 -e’ in the ellipse, = e — 1 in the hyperbola; and, if we denote the latus rectum by 2/7, we have Ly =(1 +e) m. From these formule we find the following for the ellipse, hal ie S@= ual EC=~ sla @(l- e)=—; e | a for, since AS’ = m = a (1 - e), we have SC= AC- AS=a- a(1-e)= ae, EOS AGM Osi nase Ce ‘ar =m(1l+e)=a(l-—eé) = —, ie @ The corresponding formulz in the hyperbola are | SC =- ae, TEC eu cermen b=a(e-1)=-—. ! é a Prov. XXXVII. 177. To determine the polar equation of the three curves eferred to Sas pole and SX prime radius. Using the notation and figure of Art. 165, and putting SP =r, > SX = 0, we have | QP = EM=n+m+4+rcos 6; od hence, since SP =eQP, and en = m, we find r=m(1+e)+ercos 6; F 2 68 PLANE CO-ORDINATE GEOMETRY. m (1 +e) l and therefore FS aa Niatla a ta ih he desl 2d ak 1 —ecos @ 1 —ecos 9 which is the polar equation required. If we reverse the positive direction of the angle 0, 7.e. if we put ASP = @ instead of A'SP, the polar equation becomes a d 1+ecos 9 178. Cor. When @ = - , r becomes the ordinate drawn through S; but, by (1), r=/ when 0= — Hence the latus » rectum (2/) is the double ordinate drawn through S. 179. The three curves we have been considering are of considerable importance ; we often meet with them in various — parts of mathematics, especially in Physical Astronomy. It is _ therefore necessary to be acquainted with their properties, many ; of which are very remarkable and well worthy of attention for) their own sake, independently of any use that may be made of — them. We shall devote several chapters of the present treatise to the investigation of these properties. We often meet with equations of the form Ag + Dy —C, “Ay — Be, AT =By, where A, B, C are any constants, positive or negative; and : it is important to be able to make out exactly the loci which these equations represent, and how far they depend in kind and position upon the values of the constants. On this account the following propositions are added. Prop. XX XVIII. 180. ‘To determine the locus represented by the equation — Ax’ + By’ =C. Dividing this equation by C, it becomes A B —27+—ye=l1, C C which, according as : amd % happen to be positive or negative PLANE CO-ORDINATE GEOMETRY. 69 may be put in one of the following forms, viz. x y° a y° am 3 i nt bt lye PTs ica 18 Be <0 Beye (3), se y x y? mg lo > — =~ ml... (4). (1) and (2) we have already considered; but we must observe respecting (1) that & may be greater than «, which is not con- sistent with Art. 174. We have however, in this case, only to interchange the co-ordinate axes by writing z for y and y for 2, and to assume 2 = 7 and e = * instead of 4 = = and. = = ee In this manner the equation will be reduced to the proper form. (1) therefore represents an ellipse whether b be less than a or greater than a, only in the latter case the axis of y is the axis on which the two foci are situated; fig. 60 therefore represents the locus in the latter case. (3) may be reduced to the same form as (2) by interchanging ‘the axes of co-ordinates ; therefore it represents a hyperbola having its foci on the axis of y. (See fig. 61.) (4) does not represent any locus, because all real values of 2 and y make the first side negative, whereas the second side is positive. 181. If Cbe zero we cannot divide the given equation by it, but we may determine the locus thus. ‘The equation becomes _Az’ + By’ = 0, which, according as A and B have the same or contrary signs, may be put in either of the forms 2 2 + =0 tenes (1), =~ = 0...-.- ). The first member of (1) is the sum of two essentially positive quantities, which therefore cannot together make zero, unless each be separately zero. Hence (1) gives x = 0, y =0, and these are the only values of z and y which satisfy (1). It therefore represents the isolated point (00), z.e. the origin. (2) may be put in the form mae y aca a Caer 70 PLANE CO-ORDINATE GEOMETRY. and therefore (2) is satisfied by all values of z and y which make either of these factors zero, and by no other values. Now if (zy) be any point on the right line (2 = : = ; the first factor | a | is zero, and if (xy) be any point on (: + ; = 0) the second t a factor is zero. Hence it is evident that the equation (1) repre- ‘ sents the two right lines oh = ire ny ibid wlpvabed, a Ob a Ob 182. If in Art. 165 we suppose m to be zero the equation | becomes (1-é€)2+y¥'°=0, | which is of the same form as (1) or (2) (in the present Art.) accord-_ ing ase is< or>1. We may therefore regard the point and the two right lines represented by (1) and (2) as an ellipse and a hyperbola in which the points S and A coincide. | ; Prop. XXXIX. 183. ‘To determine the locus represented by the equations Agi =. Bz, Ag =| .By. These equations, according as A and B have the same or different signs, may be put in one of the following forms, viz. JRE pect Ce Pack one ae GW) Le (Sih VD Rae AR (3), ME sin er aes (2), wa Amy... (4). (1) we have already considered. (2) may be reduced to the | same form as (1) by writing - x for z, ¢.e. by reversing the positive direction of the axis of x(see Art. 108); and therefore. (2) represents a parabola situated as in fig. 62. (3) may be | reduced to the same form as (1) by interchanging the co-ordinate axes; and (4) may be reduced to the form (8) by reversing the positive direction of the axis of y. Hence the loci (3) and (4) are parabole situated as in figs. 63 and 64. : 184. If m= 0 (1) and (2) represent the axis of z, and (3) and (4) the axis of y. If e=1 and m=0 in the equation obtained in Art. 165, it becomes y’ = 0; therefore a parabola becomes a right line when the points S and A coincide. PLANE CO-ORDINATE GEOMETRY. 74! The object of the following chapter is to shew that the loci ve have just been considering are represented by the general equation of the second degree between z and y, and that they we also identical with the curves produced by the section of I common cone by a plane in different positions. We = shall terwards investigate the various remarkable properties of the llipse, parabola, and hyperbola separately. This being the case, ie following chapter may be passed over on a first perusal of ee CHAPTER VII. OF THE GENERAL EQUATION OF THE SECOND DEGREE BETWEE ; % AND ¥, AND THE LOCI IT REPRESENTS. REDUCTION OF THE GENERAL EQUATION TO ITS SIMPLEST FORM. SECTIONS OF RIGHT CONE BY A PLANE ARE LOCI OF THE SECOND ORDER. 185. We have seen that the general equation of the first degree between z and y represents a right line; we shall noy consider the general equation of the second degree, and shew how to determine the locus it represents in any given case. The general equation of the second degree between z and contains the terms 2”, zy, y’, x, and y, each multiplied by some constant, together with a term independent of x and y 3 it may therefore be put in the form Az’ + 2Bay + Cy’ + 2Dx+ 2Ey + F = 0, some of the constants being multiplied by 2 (since we may assume arbitrary constants in any form we please), simplify certain formule we shall obtain hereafter. We shall first suppose that the term 2Bry is wanting in this equation, and on this supposition determine the nature of the locus. We shall then shew that the term 2 Bay, if it occur in the equation, may be made to disappear out of it, by simply turning the axes of co-ordinates through a certain angle ; and thus we shall reduce the general case to the particular case previously considered. in order to Prov. XL. 185. To determine the nature of the locus represented by the general equation of the second degree, supposing that the term 2Bzxy is wanting. PLANE CO-ORDINATE GEOMETRY. 73 In this case the equation is Az’ + Cy’ + 2Dxe+2Fy+F=0........ (1). Ist, Suppose that neither A nor C are zero; then we may ssume D= dh, H= Ck; and, substituting these values, (1) ecomes A (2 + 2he) + C(y? + 2hy) =- F; yr, adding Ah’ + Ck’ to each side, and putting Ah’ + Ch? - F=G or brevity, we have A(x+hy+Ciy+khy=aG; md if we transfer the origin to the point (- h, - &), which is lone by writing z-h and y-k& for z and y, this equation yecomes os Cosh 6) pA 6 cg ate ca aA C27, 2ndly, Suppose that 4=0; then, assuming H=Ck, the quation (1) may be put in the form Ciy+ky = Ce? - F-2Dz, md, assuming Ch" — I= — 2Dh (supposing D not equal to zero), ve have Ciyt+ky=-2D(e+h), vhich, if we transfer the origin to the point (— h, — k), becomes CE ae NO a Ae A MN EM GD If however D = 0, then (1) becomes Cy ero Hye Fi 0. 5h AOE, PRCA 8rdly, Suppose that C= 0; then the equation (1) may be educed, as in the previous case, to either of the forms 2a NS tial LT prea ee Re ood Aa Av’+2Dzr+F=0... Se nae a ery) The equations (2), (3), (5) we have already fully considered n Arts. 180-184. The equation (4) may be put in the form (¥-p)Y-9)= 9, which, if p and qg be real, represents (see Art. 181) the two right Ines (y = p) and (y= q) which are parallel to the axis of x ; and fp and q be impossible, it does not represent any locus. In the ame manner (6) represents either two parallel lines or no locus. Hence it appears that the general equation of the second legree wanting the term 2 Bry represents an ellipse, a parabola, 74 PLANE CO-ORDINATE GEOMETRY. . a hyperbola, two right lines, one right line, or an isolated point; except in certain cases where it does not represent any locus. The following proposition will complete the discussion of the general aphttans ee Prop. XLE 187. To shew that the term 2Bzry may be made to disappall ; from the general equation of the second degree by turning the axes of co-ordinates through a certain angle. If, in the general equation 7 Ax’ + 2Bay + Cy’ + 2Dx+2Fy+ F=0...,(1),9 we write x cos 6 — y sin @ in place of z, and z sin 0 + y cos 0 in place of y, we turn the axes through the angle 6, leaving the origin unaltered (see Art. 104). When these substitutions for x and y are made, (1) may evidently be expanded and arranged so as to assume the form | A'z? + Bry + Cy? + 2Diz + 2E'y + F=0....(2), 0 A’, B', C’, D’, &c. being quantities formed from A, B, C, D, and @, in arranging the equation in the form (2). rm hy 2B'zy is the only term of (2) we have occasion to know for our present purpose ; it can only arise from the three first ter ns of (1), which, after the substitutions for z and y, become A (x cos 9 - y sin 6) + 2B (x cos 6 - y sin 9) (x sin 0 + y cos 0) + C(x sin 0+ y cos OY, or A'z*+2{—A cos 0 sin 6 +B(cos’6 - sin’0)+C sin @ cos O} zy oy Hence 2B’ = 2B cos 20 —-(A — C) sin 20. | Now, from this expression, it appears that we may mall B' = 0, by putting * i > tan 20 = ee Sele we Leak oe A-C a and this we may always do, since the tangent of an angle may be of any magnitude, finite or infinite, positive or negative. Hence it appears that, when 8 is determined from (3), the term 2B'zy does not occur in (2). Consequently, by turning the axes of co-ordinates through a certain angle, determined by (3), we may always make the term 2Bzy disappear from the general equation of the second degree. Q. 5. D. PLANE CO-ORDINATE GEOMETRY. 75 188. Cor. Hence, and by the previous proposition, the ost general equation of the second degree represents an ellipse, ; parabola, a hyperbola, two right lines, one right line, an lolated point, or no locus. 189. By means of this and the previous proposition we may stermine the form and position of the locus represented by any ven equation of the second degree. We must first get rid of ie term 2Bery, if it occur in the equation, by determining the roper value of 6 from (3), and substituting it in 4’, C’, D', and "; and then we have only to reduce the resulting equation ; in Prop. xu., and so find the locus required. But this rocess is rather tedious, and we may determine the locus much ore readily by means of the following proposition. Prop. XLII. 190. If the equation Ax’ + 2Bry + Cy’ + 2Dx+2By+ F=0....(1) 2 reduced to the form | Ag+ Oy +2D2+2hy+ F=0......%.(2), y turning the axes of co-ordinates through an angle 9; to stermine A’, C", D', EL’, and 0. Since (1) is reduced to (2) by turning the axes through the igle 0, it follows that (2) is reduced to (1) by turning the axes » which (2) is referred through the angle —- @. Therefore, if e put 2 cos (— 9) - y sin (— 9) for z, and x sin (- 0) + y cos (— 8) ry in (2), it must become identical with (1). Making these ibstitutions in (2), it becomes "@cos 6 +y sin 0+ C'(- «sin 0+y cos 0)’ +2 D(z cos 6+ y sin 8) + 2h'(- x sin 8+ y cos #)+ F=0. ence, equating the coefficients of 2°, zy, y’, x andy in this {uation to those in (1), we have FAR COR g URC ar Siti Gy Ae Pees st (3), PAS ae O COn De Pre sg (4), Alsi OO cps Dim OG. oc sacs acs 8 (5), D cos @- E'sn#=D........ poi, GOs PP Ri ok BE) CORO Fis ate ia asm wih) 76 PLANE CO-ORDINATE GEOMETRY. The first three of these equations determine 4’, C’, and 6 and then the two last give D' and HE’; and thus we obtain the five required quantities. 191. A few examples will shew the use of this proposition, Ex. 1. To determine the locus represented by the equation — bx? + Qay + 5y® - 12cV2y — 12eV2z = 0. A In this case the above equations become A».cos’ 0.4) gin’ 035 ee eee (1), | (A"— GD¥sin 0icos,0;=slashs oye coe (2), A’ sin’@:+.:C" cost: Hie 5a, Soke aie (3), D' cos 0 - E' sin 0 = — 6ev2....... (4) D' sin 0+ £E' cos @=- 6ev2....... (5) (1) — (8) gives (A' — C") cos 20 = 0, 1 which, since A’ -— C" is not zero by (2), gives 0 = 45°; and therefore (1) and (2) become AOC 10, say Usa and therefore A Ba @ Mo a 64: Also (4) and (5) become Di eo q Di 2ST eee oe! . and therefore Da — 1 26h a: ‘] Hence, by turning the axes through an angle 45°, the given equation becomes 62° + 4y* — 24cy = 0, or 3xu° + 2y - bef = 72¢*, which, if we transfer the origin to the point (0, 6c), becomes — 2 2 240° = 36” and this is the equation of an ellipse whose axes are 2 V6 and 6c. Hence, if we take X'OX = 45° (fig. 65), draw OC (= 66) at right angles to OX', and CA parallel to OX’; then th ellipse described on C'A and CB as semiaxes is that represented by the given equation, where CA = 2c/6, CB = 6c. PLANE CO-ORDINATE GEOMETRY. PT 192. Ex. 2. Let the given equation be x + Qry — y® — 2cx + Acy — 4c’ = 0. In this case we have Pag COS Uae Wine Osh ii see ihre ok Ly; (Ae Carsineiieos) 0 ale eis. ts. <3 (2), ‘Alvain Gare Aoos, Oy Ti genni ees: 4.08); “ |, becomes! Ga er, Hence the given equation represents a parabola. 194. Ex. 4. Let the given equation be “x - 2tzy+y° - 2c =0. Here 0= 45°, A’=0, C=2, D=0, H=0; therefore by turning the axes through an angle 45°, the equation becomes y—-c=0, or (y-e)(y+e)=0, which represents two parallel lines. Indeed this may be seen) immediately, for the given equation may be put in the form (4 -—y - cv2)(x«-y+ev2)=0, | which represents two parallel right lines making an angle 465 with the axis of z. Prov, XLII. 195. To shew that the general equation of the second degree represents, in gener al, an ellipse, parabola, or hyperbola, according as dC’ ~ B? is positive, zero, or negative. Tt is evident, from Art. 186, that the equation (2) in Art. 19 ) represents, in general, an ellipse, if A’ and C’ have the sam sign; a peehal if either A’ or C’ be zero, and a hyperbolai A’ and. C" have opposite signs; ze. the locus is an ellipse, parabola, or hyperbola, according as A’C" is positive, zero, oF | negative. Now, referring to the equations for determining” 0, A’, C’, &c. in Art. 190, we may thus determine 4’C’. (3) x (5) gives (A® + C™) sin’ 0 cos’ @ + A'C" (cos* 0 + sin‘ 0) = AC... (8) and (8) — (4) gives A'C" (2 sin’ 6 cos’ 9 + cos‘ 9 + sint 9) = AC — B’, or AOAC = PLANE CO-ORDINATE GEOMETRY. 79 ‘Hence it follows that .4’C’ is positive, zero, or negative, cording as AC’— B* is so; and therefore the truth of the MBoposition is manifest. | Prop. XLIV. 196. To determine the general values of 0, A’, C’, D', E' _ from the equations obtained in Art. 190. 3) — (5) gives CANS CorcosaG 274 iO 2) ee (9), aod 2(4) + (9) gives 9B tan 20 = - pe Sn vy Bhich ; is the same value of tan 20 as that found in Art. 187. _ Again we find, as in the preceding article, ba ACh = AC Bp’. A Also (3) + (5) gives A’'+ (’=AiC. _ Hence A’ and C’ are the two values of z given by the quad- atic equation Pe (A + Cyz od Ce Re | - Lastly, the values of D’ and E' obtained from the equations '(6) and (7), are ae D=Dcos6+ Hsin, EF’ = Ecos 0- Dsin 0. _ Thus the values of 0, 4’, C’, D', and E’, are completely | — \ : | 1 Prop. XLIV. (é2s.) 192 (dts). ‘To cause the terms 2Dz, 2/y to disappear from ‘the Be eeral equation 5 Az’ + 2Bry + Cy’ + 2Dxe+ 2Hy+ F=0...... CE); ' ‘Transfer the origin to any point (A/), and the equation Beh) 2B(a+h)(ytkh)+Cy+ky+2D (e+h)+2h(y+h)+F=0, Ae aBiey Cr +2(Ah+ Bk+D)x+2(Ch+ Bh+ L)y+ F'=0...(2), vhere F'= Al? + 2Bhk+ Ck +2Dh+2Ek+ F. fi Now if we assume, as we may do, | Ah+ BE+ D=0....(8), Chi Bh+ E=0,...(4), (which two equations Aeccnine hk and 4), the eraeee (2) be comes ae erie Cys te Eee Os. a ws 05 gas (5). 80 PLANE CO-ORDINATE GEOMETRY. : Thus we have made the terms 2Dz and 2Ey to disappear by. transferring the origin to the point (2A), 2 and & being given by. the equations (8) and (4). t 193 (dts). Cor. 1. (8)h +(4)& gives Al? + 2Bhk + Ch? + Dh + Ek =0; therefore the value of F” becomes Dh + Ek+F. Hence it appears that, if we transfer the origin to the point (hh) deter- mined by the equations (3) and (4), the equation (1) becomes — Az’ + 2Bey + Cy’ + Dh+ Ek+ F=0...... (6). 194 (dis). Cor. 2. The values of 4 and &, given by (8) and. (4), are Pa! DC- EB ae Og ipa re isu: ACs OB & TAG pe Hence, by Art. 195, and & are infinite in the case of the parabola. 195 (d¢s). Cor. 3. Ifin the equation (6) we put z =7 cos 0, y =r sin 0, we evidently find r=+vVP (where P is a quantity” depending upon 9, the precise value of which we have no occa- sion to know). Hence every value of @ gives two values of re one the negative of the other ; and ther Site every chord drawn through the origin is bisected at the origin. It appears there- | fore that the curve is perfectly symmetrical with respect to the | origin, and consequently the origin must be its centre. Hene@l Hie) Boe (hk), determined by the equations (3) and (4), is the centre of the locus represented by (1). | 196 (dzs). In those cases where AC - B’ is not zero, cha present proposition will enable us to simplify the application of the method given in Prop. XLII.: for, by first reducing the general equation to the form Az’ + 2Bzry + Cy’? + I" =0, and then applying the method alluded to, we shall have no occasion” to employ the equations (6) and (7), (Art. 190), since D’ and BE will evidently be each zero. For example, let us take the equation in Art. 192, namely a’ + 2ey - y’ — 2cx + 2cy — 42° = 0. | i If we transfer this to its centre, it becomes, by equations (3), (4), (6) in the present Prop., z+ 2zy-y'-ch+ckh- 4=0........(7), PLANE CO-ORDINATE GEOMETRY. 81 At+k-c=0, -k+h+c=0; and therefore h= 0, k =c; which being substituted in (7), it becomes x + Qxy — y’ — 8c? = 0. Now, proceeding exactly as in Art. 192, observing that D and # and therefore D' and E' are zero, we find SWI IOS iW; 9° “ and therefore ey’ = asks : V2 which i is the same equation as that obtained in Art. 192. We have here determined the co-ordinates of the centre to be 0 and ¢, whereas in Art. 192 we only obtained them in terms of 0 (which = 223°), In cases where 4C'— B* = 0, h and & are infinite (in general), and we must therefore proceed altogether as in Prop. xuit. : Of the Sections of a Right Cone by a Plane. | The ellipse, parabola, and hyperbola, are aften called lines of ‘the second order, because they are represented, as we have seen, by the general equation of the second degree between z and y. But they are more commonly known by the name of conic sections, because they are the curves produced by the section of a cone by a plane, as we now proceed to shew. Prop. XLV. 197. To shew that the section of the surface of a right cone made by any plane is always a line of the second order. A right cone is the solid generated by the revolution of a right-angled triangle round one of its sides. By the surface of this cone we mean the surface generated by the revolution of the hypotenuse produced indefinitely both ways. The side round which the triangle turns is called the axis of the cone. Tf any plane be supposed to cut the cone, the curve in which the conical surface meets that plane is termed the section of the conical surface made by it. When the cutting plane is per pen- dicular to the axis of the cone, it is evident that the section is G 82 PLANE CO-ORDINATE GEOMETRY. a circle, its centre being the point where the axis meets the plane. Let ABC (fig. 66) represent the cone, and OPQP’ the: section made by the cutting plane ; let RPSP' be the circular section made by any plane perpendicular to the axis, and let ARMQ be the plane containing the axis of the cone, and per- pendicular to PMP, the line of section of the two former planes ; then PM is perpendicular to OM and #M, and RS is: a diameter of the circle RPSP’. : Take OQ as axis of z, and let OM(= x) and MP(=y) be the co-ordinates of P, which may be any point of the section OPQP'; then, since PM’ = RM. MS by a property of the circle, we have fi RM. US... eee (1). 9 Now, drawing ON and 7'Q (fig. 67) parallel to RS, we have RM OM MS MQ (2). TQ~ 00’ ON 0Q Hence, if we put ON=c, ROQ=0, OAQ = 2a, OQ = 2a, and therefore ; j G sn TOQ _ sin j eh ele sn ATQ cosa’ ; we have, by (2), since OM = z, and MQ = 2a -2z, q RM: US = creo ga i 2a cosa } Hence (1) becomes ) e sin 8 ‘ 7 = — 8G 2 Nee ee i J 2a cosa Sr ee (3), ‘ which is an equation of the second degree between # and y. and therefore the section of a right cone by a plane is always é line of the second order. Q.E.D. ; j 198. Cor. 1. Since = = ie , (3) may be put ix a the form { »_ sin 0 {or - sin (0 — 2a) el. WE OR a COS a ; ; or Ay’ + Bx’ - Cx = 0, j when A=cos?a, B=sin @sin(@-2a), C=csin 6 cosa. PLANE CO-ORDINATE GEOMETRY. 83 md if it fall on the left, 6 is < 2a; and in these cases B is ero, positive, or negative, respectively, Hence, since A is Peis positive, the curve 1s an ee parabola, or mt CHAPTER VIIL. } r OF THE PARABOLA. PROPERTIES OF THE PARABOLA NOT CONNECTED WITH TANGENTS OR DIAMETERS. PROPERTIES CONNECTED WITH TANGENTS. PROPERTIES CONNECTED WITH DIAMETERS. VARIOUS PROBLEMS. 7 200. We now proceed to investigate in detail the various remarkable properties of the ellipse, parabola, and hyperbola. We shall consider each curve separately, commencing with the parabola, because it is the simplest of the three. 4 201. The equation of the parabola referred to its vertex A as origin is, by Chap. v1., : f 4)” = AINE el atone eee ke ae (A): m denoting the distance of the focus (S) from the vertex. Since e=1 and AS =e. EA, mis also the distance of the foot of the directrix from the vertex. The general value of the latug rectum is 2m qd +e), therefore, putting e=1, the value of the latus rectum in the parabola 1 is 4m. Hence, in the parabola, the square of the ordinate is a mean proportional between the abscissa and latus rectum. The polar equation of the parabola referred to S as pole # is, putting e = 1 in Art. 177, k 2m _ a ee ORS 6 800s O28 160. e 1 ene 2 e 5) 1 — cos 8 (2) | From these equations we shall investigate the various remarkable properties of the parabola. é ; + bd PLANE CO-ORDINATE GEOMETRY. 85 Properties of the Parabola not connected with Tangents or Diameters. Prop. XLVI. 202. The distance (SP) of any point (zy) of the parabola from the focus (S’) is equal to z + m. For, SP being the distance between the points (m, 0) and (zy), we have SP? = («- my +y? =2 + 2mx+m’*, since y® = 4mz; therefore Ni Dee 6 O, Hed. 203. Cor. If we turn the axes round through any angle @, and transfer the origin to any point (hk); in which case we must, by Art. 106, write z cos 0- y sin 0 +h for x; we find SP =x cos 0-y sin8+h+m. Hence, to whatever axes of co-ordinates the parabola be referred, the expression for SP is always a linear function ‘Art. 50, Note) of the co-ordinates of P, i.e. an expression of the form Az + By +O. Prop. XLVII. 204, To determine in general the locus of a point (xy), Whose distance from a given point (4&) is a linear function of rand y. Let the expression for the distance of (xy) from (hk) be Az+ By+Cer. _ Now the formula for the perpendicular distance of the point zy) from the right line whose equation is Agar Dyin Crea Vara wscrcen aa (1), Age Dye Cee s, by Art. 69, Wa BT P; nd therefore r= /(A* + B’).p. * This may be also shewn very readily by the polar equation, and geome- rically, as follows. In fig. 68, QP being parallel to 4S, we have SP=QP=AM+ FA=x+m, since FA=m. 86 PLANE CO-ORDINATE GEOMETRY. Hence the perpendicular distance of (zy) from the line (1) is | always proportional to its distance from the point (AA). There: fore, by Art. 165, the locus of (zy) is a conic section, whose: focus is (hk), whose directrix is (1), and whose eccentricity is /(A® + B’). 205. Cor. 1. Hence we may define the focus of a conic section to be that point whose distance from any point (xy) of the curve is a linear function of # and y. 206. Cor. 2. Hence, if the conic section be a parabola referred to its axis as axis of w and its vertex as origin, (1) being the equation of the directrix, gives B= 0, ae m; an d, since the eccentricity is unity, we have A’ + B’ or A’ equal to 1. Hence r=xr+M, which is the result of Art. 202. Prop. XLVIILI. ' 207. If PSP’ be any chord drawn through the focus, the the semi-latus-rectum is a harmonic mean between SP and SP. For if PSX = 6 we have, by the polar equation, 1 1 —cos 0 “i SB amma 4 and putting 0 +m for 9, and therefore SP’ for SP, we have i 1 1+ cos 0 SP i eee rns Hence 2 reer Kijak Wee which shews that Zis a harmonic mean between SP and SP’. — Prop. X LIX. 208. ‘Two given lines being drawn through the vertex of parabola, to find the equation of the line joining the other tw points in which they intersect the curve. Let the equations of the two lines be y—azr= 0, y-ar=0. ‘ , . . PLANE CO-ORDINATE GEOMETRY. 87 Then, as in Art. 181, both the lines together are represented by the equation (y — ax) (y - a'z) = 0, or y” + aay” — (a +a’) zy = 0. Where this locus intersects the parabola we have y’ = 4mz ; therefore, substituting this value for y’ and dividing out z, we have 4m +aag—(a+a')Yy we de... see ely; which equation is satisfied by the co-ordinates of both the points where the two lines again meet the parabola; therefore, reason- ‘ing exactly as in Art. 139, (1) is the equation of the right line joining these points. 209. Cor. Putting y = 0 in (1), we find z = - alk , which, aa if the two lines be at right angles, and therefore aa = — 1, becomes 2 = 4m. Hence, if any two lines be drawn from the vertex of a parabola at right angles to each other, the line join- ing the two points where they again meet the parabola always - crosses the axis at a distance 4m from the vertex. Prop. 210. To determine the length (7) of a right line drawn, at an angle @ to the axis of z, from any point (1A) to the parabola. Proceeding exactly as in Arts. 68,128, putting h +r cos 0 and k +7 sin @ for 2 and y in the equation of the parabola, we shave (k +r sin 0Y = 4m(h +r cos 8), or sin? 0 — 2(2m cos 0- k sin 0)r +k - 4mh = 0...(1), which determines the required distance. Since this is a quadratic equation, the right line must in general meet the parabola in two points whose distances from (hk) are the roots of (1). 211. Cor. 1. Using the same notation as in Arts. 129, 130, we have 2m cos 6 —& sin 0 PQ+PQ =2 eee eo sin’ @ kh? — 4mh sin? @ Core. 0) ee PORP O= s 4 88 PLANE CO-ORDINATE GEOMETRY. | 212. Cor. 2. Hence, as in Art. 182, if (hk) be the middle | point of QQ’, we have 9m cos 0- ksin 9=0.....5. 005. sae | which is therefore the condition necessary in order that the - point (hk) may be the middle point of a chord which makes an | angle @ with the axis of z. Prop. LI. | 213. If QQ’, RR’ be two chords of a parabola, and P thei point of intersection, the ratio PQ. PQ': PR. PR’ is not altered by moving each chord parallel to itself, and so shifting the position of P in any manner. By the previous Article we have ane PQ.PQ'= alla sin* 0 | 6 being the angle which QQ’ makes with the axis, and (hk) the | point P. In the same way, if ¢ be the angle which RR’ makes with the axis, we have PR PR eee sin’ ® PQ.PQ sin’¢ PREP RIS pa Now if we move each chord parallel to itself im any manner we do not alter 6 or @; hence the truth of the proposition is | manifest. therefore Cor. If either 0 or @ be equal to 0 or 7, we must state this proposition somewhat differently. Suppose that @ = 0, then the equation (1) becomes —~4mr+kh —-4mh= 0, which shews that the right lme meets the parabola in only one point, Q suppose, and that PQ= Therefore, by (5), we have PR.PR 4m PR sin’ p k? — 4mh Amn ts PLANE CO-ORDINATE GEOMETRY. 89 Hence, if from any point P of a chord RR’ we draw the line ?Q to the parabola parallel to the axis, the ratio PR.PR': PQ 3 not altered by moving PQ or RA’ parallel to itself in any aanner. _ Tke remarkable property of two intersecting chords proved ‘a this proposition is true for all the conic sections. Properties of the Parabola connected with Tangents. Prop. LII. 214. To determine the equation of the right line which ouches a parabola at a given point. Proceeding exactly as in Arts. 135, 136, if we suppose the ight line in Art. 210 to move parallel to itself till Q and Q’ soincide, and if we also suppose P to coincide with Q, the equation (2), Art. 211, becomes = 2m cos 0-—£ sin 8, or tarts (ete FPN) tee arenes (1), yhich is the tangent of the angle which the line touching the varabola at the point (h/) makes with the axis. Hence the equation of this line, since it passes through (4), ‘s k(y —k)- 2m(« -h)=0, which, since /* = 4mh, becomes | Ree (Arnie le Ccrsete te a enrree tae (2) and this is the equation required.* * We may obtain this result somewhat differently in the following manner. Let (2k), (i’k’) be any two points on the parabola; then the equation of she line drawn through these ponte is y—-k= de (eR) ale eke (4), Now, subtracting the equations h* = 4mh, k? = 4mh', we find k”-F =4m (h'—h), and therefore ee rot Zi ; hence the equation becomes | —h y-k= = ary Ga h); pee 90 PLANE CO-ORDINATE GEOMETRY. Prop. LILI. 215. If y=az+ B be the equation of any tangent. of th | parabola, to determine what relation must hold between a and [3. Let (hk) be the point of contact, an by Art. 214, the two equations y=ar+, and y= (2 +h), must represent the same line, and therefore we have 2m 2mh _ i; =a, —_ = se 7. > therefore Gh =m, since hk’ = 4mh. Hence af = mis 4 relation required.* + a 4 which equation may be made to differ as little as we please from the equations g-k= om fa-Bys mo he, (5), by bringing the point (///’) ae near to the point (hk). Hence the line represented by (4) may be made to approach as near as we please to the line represented by (5), by making (Wk') approach (kk); which eviden y cannot be, unless (5) be the tangent at the point (hk). (5) therefore repre- sents the tangent at (44), and it may be reduced, as in the text, to the form — ky =2m (x +h). It is erroneous to say that the line (4) becomes the line (5) when i) coincides with (2k), for two reasons: Ist, when (/’'A’) and (hk) coincide, (4) is no longer a definite line, but any line whatever drawn through (U5 2ndly, we are not at liberty to ni ide the equation k” — Kk? —-4m (h' —h)=0, | by h’-h when h'=h; for ie is no rule of Algebra which enables us to. Ki-k 4m hohe K'+k P| divide an equation by zero; and therefore we cannot prove that when (7’k') and (hk) coincide. * We may prove this result somewhat differently, as follows. | If we eliminate y between the equations y= ax +f and Je =4mx, we find | (ax+P)’=4mx, or az?4+2(aB-Qm) 2+? = Now, if the line touch the parabola, this equation, which determines the : abscisse of their points of intersection, ought to give two equal values of 2 and therefore its first member ought to be a perfect square ; we have therefor a®? = (a8—2Qm), or aB=m. : PLANE CO-ORDINATE GEOMETRY. 91 Hence the equation of a line touching a parabola (in terms simply of the angle it makes with the axis) is m b y= an + — A Piece epee are ate ei Gs OF Prop. LIV. 216. To determine the subtangent and subnormal of any point of the parabola. If T (fig. 69) be the point where the tangent at P meets the axis, the line 7M is called the swbtangent; and if PG be perpendicular to PT, PG is called the normal, and MG the -subnormal. The point P being (AX) the equation of PT is ky — 2m (# + hy=O wrcecseeeees Gh}, and, PG being a line drawn through (4/) at right angles to this line, its equation is . am(y—kh)t+k(e@-h)=0....cceeee 2: And, if we put y = 0, x becomes AT in (1), and AG in (2). Hence AT=-h, or TA=h, and AG =2m +h; and therefore TM = TA + AM=2h, MG = AG —- AM = 2m. Hence the subtangent of P is twice the abscissa (AM), and the subnormal twice the distance AS. 917. Cor. TS = TA+ AS=h+m; but, by Art. 202, SP =m+h; hence TS = SP, therefore the triangle TSP 1s an isosceles triangle, and the angle 7'PS is equal to the angle PTS. Hence the tangent at P makes equal angles with SP and the axis; or, if we draw PN parallel to AS, the tangent PT bisects the angle SPT* * We may shew this geometrically, as follows. Let P’ (fig. 70) be any point of the parabola taken very near P, draw PQ, P'Q parallel to AS, PV perpendicular to QP, and take SU=SP. Then, since SP = PQ and SP' = P'Q, we have VP'= UP’; also, since 7 _SP=SU, and U may be taken as near P as we please by making P’ approach P, itis evident that the angles at U may be made to differ from right angles as little as we please. Hence in the triangles PUPRD PREY; we have PP’ common, P'V = P'U, and P' UP may be made as nearly equal 92 PLANE CO-ORDINATE GEOMETRY. 218. This may be also shewn as follows. The equation of Z'’P being ky — 2m (x + h) = 0, we have 2m tan 7’ = Tae 9 2m k 4mk k and therefore tan 27’ = pom Yamorrny wearers =r bere ke Now the equation of SP, a line drawn through (m, 0) and | (hk), is i ORNer Tre | FO ae Sg > and therefore tan PSX = = tan 27. tad Hence PSX = 2PTS, and therefore PTS = SPT. i Prop. LY. 219. To find the locus of the intersection of a tangent with the perpendicular let fall upon it from S. By Art. 215, the equation of the tangent is and the equation of a line at right angles to this through Sis 1 fete ool Bere 5 A Ne tes) Suppose x and y to have the same values in (1) and (2), and therefore to belong to the point of intersection of the two lines ; then (1) — (2) gives 7a a+ =) z=0, a . . i . . . which, since a + ~ cannot be zero in general (nor indeed in any a case) gives me 7 3 to P'VP as we please, by making P’ approach P; therefore PP'U may be made as nearly equal to PP’V as we please, by making P’ approach P; which cannot be, unless the tangent at P makes equal angles with PS and PQ. ng PLANE CO-ORDINATE GEOMETRY. 93 which, being true for all values of a, shews that the point of intersection of the two lines is always on the axis of y. Hence the axis of y is the locus required.* 220. Cor.1. This proposition suggests the following method of drawing (geometrically) a tangent, or rather two tangents, to a parabola from any given point U (fig. 72). Join S and U, on SU as diameter describe a circle cutting AY at Rand R’; then the lines drawn from U through # and F will touch the parabola. This is evident, smce SRU and SR'U are right angles; and therefore, by the proposition just proved, UR and UR' produced jaaust be tangents to the parabola. 221. Cor. 2. If U be on the directrix, the point V where SU meets AY must be the middle point of SU; therefore RR’ must be a diameter of the circle, and consequently RUR' must ‘be aright angle. Hence the two tangents drawn from any point lof the directrix must be at right angles to each other. This we ‘shall prove analytically in the following proposition. Prop. LVI. _ 222. ‘To determine, analytically, the tangents of a parabola which pass through a given point. : { The equation of the tangent is 7 a Suppose it to pass through a given point (A/), then we have mM or a—-—a+t * This may be shewn geometrically, as follows (fig. 71). + Let PR be the tangent at P meeting QS at FR; then, since PS= PQ, | and QPR=SPR, PR must be at right angles to SR; also we have | SR= RQ, and therefore, since S4= AH, AR must be parallel to HQ, ve. | AR is the axis of y. Hence the tangent and perpendicular from S$ upon it “intersect in the axis of y. | ' j | | a 94 PLANE CO-ORDINATE GEOMETRY. which equation gives two values of a, and therefore shews thai in general two tangents may be drawn through a given point, If a, a’ be the two roots of (1), the equations of the two tangents are m ’ m 1) a OO Fos ther eo eveoee (2), Lo Coane oe ede a 223. Cor. To determine the locus of the point of inter- section of two tangents at right angles to each other. : If (2) and (8) be at right angles, we have aa +1=0. Pika But, by (1), aa’ = sat hence gO aRNar neers h which shews that the point of intersection of the two tangents is | always on the directrix. ‘The. directrix is therefore the locus” required. Prop. LVII. 224. ‘The tangent at P, and the right line drawn through § 7 at right angles to SP, meet the directrix at the same point. Supposing (h/) to be P, the equation of the tangent is ky -2m(x+h)=0; 3 and therefore, putting z = — m, we find that the ordinate of the | point where the tangent meets the directrix is q 2m (h - m) k ¢ Now ; £ ss is evidently the tangent of the angle which SP makes with the axis, and therefore the equation of the line drawn through S at right angles to SP is Tesolin ea 162* 5% and therefore, putting z = — m, we find that the ordinate of the point where this line meets the directrix is 2m (h — m) ali eee Hence the two lines meet the directrix at the same point. PLANE CO-ORDINATE GEOMETRY. 95 | Prov. LVIII. _ 225. To find the length of the perpendicular upon the tan- - gent from the focus. f _ The equation of the tangent being a ky = 2m (x + k), the length of the perpendicular upon it from the point (m, 0) is, by Art. 69, 2m (m + h) | V(hP + 4m’*)- Hence, if we represent this length by p, and observe that 7 =4mh, andm+h=SP =r1r, seep ge we have | p =mr," | Ww hich gives the perpendicular upon the tangent in terms of the radius vector of the point of contact. Prop. LIX. 226. ‘To determine the equation of the line joming the Bis oints of contact of the two tangents drawn from a given point to a parabola. j q 4 Let (AA) be the given point, and (zy) either of the points of contact ; then the equation of the tangent gives ky — 2m(«#+h)=0; 2 d, reasoning just as in Art. 139, we may shew that this is the ~ equation of the line joining the two points of contact. Q.#. F. _ Cor. 1. Hence, reasoning exactly as in Art. 140, we may shew that the locus of the intersection of the two tangents drawn at the extremities of any chord passing through the point (AA) is : ky — 2m (#+h)=0. _ 227. Cor. 2. Hence it appears that the equation, F i ky — 2m (a + h) = 0, represents three very different lines, namely, 1st, the tangent at i =. point (Ak); 2ndly, the line joining the points of contact of _ * This result is easily proved geometrically, thus: In fig. 71, since SRP=90°, RAS=90°, we have eS SP: SR=SR:SA, and «. SP.SA=SR’, Jy which gives p> = mr. 96 PLANE CO-ORDINATE GEOMETRY. the two tangents drawn from the point (A/); and 8rdly, the locus of the intersection of the two tangents drawn at the extre mities of a chord which always passes through the point (24). 228. Cor. 8. Hence the locus of the intersection of the two tangents drawn at the extremities of a chord, which always passes through the focus, is the directrix. For, putting / = 0, h =m in the equation ky — 2m (z+ h)= 0, it becomes # = — m, © which represents the directrix. 229. Cor. 4. Hence, by Art. 228, the two tangents at the } extremities of a focal chord are at right angles to each other. __ Properties of the Parabola connected with Diameters. Prop. LX. 230. To determine the diameter of a given system of — parallel chords in the parabola, ¢.e. the locus of the middle — points of the chords. Let 6 be the angle which any chord makes with the axis, and (zy) its middle point; then, by Art. 212, we have < 2m cos 9- y sin 0= 0; and consequently, if we suppose the chord to move parallel to itself, and therefore 9 to be invariable, this equation represents _ the locus of its middle point. Hence the diameter of a system. ? of parallel chords, inclined at an angle 0 to the axis, is the right line whose equation is y = 2m cot 8, which represents a line parallel to the axis. 231. Cor. If we suppose the chord to move parallel to — itself until its extremities coincide, it then becomes a tangent ; | and hence it follows that the tangent at the extremity of any diameter is parallel to the chords of that diameter. This may — also be easily seen from the equations of the diameter and tan-_ gent; for if (hx) be the extremity of any diameter, and 6 the angle its chords make with the axis, we have, by the equation of | the diameter just obtained, PLANE CO-ORDINATE GEOMETRY. 97 hence, by the equation of the tangent (Art. 214), @ is also the angle which the tangent at (44) makes with the axis; therefore the tangent is parallel to the chords. Prop. LXI. 232. To find the equation of the parabola referred to any diameter as axis of x, and the tangent at its extremity as axis of y. Let QX’, TQ Y' (fig. 73) be any diameter and the tangent at its extremity, RP any chord parallel to QY’, and therefore, by the preceding Article, a chord of the diameter Q.X’, and con- sequently bisected by QX’ at P. Then, by Art. 213, we have Pir hee eh BBO i eT’ ¢ TQ’ TA : ; or PR . PQ, since PR= PR. Hence, if we put 2 TQ’ PQ= = 2, Pha y,aand 4 = 4m’, for brevity, we find TE as Re i BPO eg ACRE OE which, being a general relation between the co-ordinates of any _ * This result may be obtained by transformation of co-ordinates, in the following manner. If in the equation y*’=4mz (which is referred to 4X, AY) we put for z, h+x+y cos @, and for y, k+y sin 0, we have : (A+ y sin 9)*=4m(h+x+ycos6).... (2); and this substitution amounts to transferring the origin to (24), and inclining the axis of y so that it may make an angle 6 with the axis of z, (this appears easily from fig. 74, where 477 and MR are the old x andy, and QP, PR thenew; AN=h, NQ=k, RPX'=9). (See Art. 110.) __ Now, supposing (2%) to be the extremity of any diameter, and 8 the angle which its tangent makes with 4X, we have h?=4mh, and k sin §= 2m cos 0; hence (2) becomes : 4 Ua dda z=4m'x, putting a = 4m. sin? 0 sin? 0 4m Kk? Also, — =4m (1+ cot? 0)=4m| 1+ — )=4(m+h)=SQ. sin’ @ 4 H 98 PLANE CO-ORDINATE GEOMETRY. point of the parabola referred to QX’, QY’, is the “a required. Let AM (= h) MQ (=k) be the co-ordinates of Q referred to AX, AY; then, since TA = AM = h (by Art. 216), we have TQ? 4 +k Ey en =h+m,_ since k* = 4mh = SQ, by Art. 202. Hence TG. It appears therefore that the equation of the parabola referred to QX’ and QY’ is of exactly the same form as the equation referred to 4X and AY, and that the coefficient of z in both equations is four times the distance of the origin from the focus. The quantity 4m’ is sometimes called the parameter of the diameter QX’.* Prop. LXII. 233. A diameter and its tangent being taken as co-ordinate axes, to find the equation of the line touching the parabola at a given point (24). : Proceeding exactly as in the note to Art. 214, we may shew that the equation of the tangent required is : ky - 2m' (a +h)=0. 234. Cor. 1. Hence the subtangent is, as before, twice thal abscissa. 235. Cor. 2. Hence, also, the equation of the line joining the points of contact of two tangents drawn from any point (2A) is, as before, hy — 2m’ (x + h) = 0. | 236. Cor. 3. If in this equation we put y= 0, we find « =—h, and this is true whatever & be; hence, if QM (fig. 75)’ =h, and we take QN = — A, and draw NR parallel to QY’; the line joining the points of contact of two tangents drawn from any point of VR always passes through M@. (See Art. 226.) * Suppose the double ordinate RR’ to pass through the focus S; then, PQTS being a parallelogram, we have PQ=ST=SP (by Art. 217) =m’; hence by (1) we find (since x=PQ) y?=4m”, and therefore Qy = 4m’, Hence it appears that 4m’ is equal to the double ordinate passing through the focus; z.e. the double ordinate which passes through the focus is equal to the parameter of the diameter of that ordinate. PLANE CO-ORDINATE GEOMETRY. 99 237. Cor. 4. If we put & = 0, the equation becomes z = ~ h, which represents a line parallel to QY'’. Hence the line joining the points of contact of two tangents drawn from any point of 1 diameter (produced of course) is parallel to the tangent at the extremity of that diameter, and is therefore bisected by that liameter. This may be also seen immediately in the following manner. Let two tangents be drawn at the extremities of any double wrdinate of a diameter, then the two subtangents (the diameter deing axis of x) are equal, since they are each twice the abscissa vf the double ordinate ; and therefore the two tangents meet the ameter in the same point. Therefore, &c. Various Problems connected with the Parabola. #38. A parabola being traced upon paper, to determine its xis. Let fig. 76 be the given parabola; draw any two parallel thords PP’ QQ’, bisect them in M, N, draw RR’ perpendicular o MN, bisect RR in B, and draw BA parallel to MN; then 4B is the axis required. #39. , z) , gives 2 p+t d' = 2a = ——= =p+4m. ay "I PLANE CO-ORDINATE GEOMETRY. 101 Hence d+d'=p+4q. 244. If P, P’ be the points of contact of two tangents drawn from a point 7’ to a parabola, and if 7PP’ = 90°; then T'P is bisected by the directrix. Let the co-ordinates of 7 be hk, and of P, h’k'; then the , equations of 7'P and PP’ are DEP TT A CACM) Rai ahaa pate ayant abe } FeAf AU Ea DE ee On eis Cx oievapare atk (2); and therefore, since these lines are at right angles, we have : = age adler ene = ; LR Pee OM Kye ge oa OMe ee ca, (3); also, since (h/) is a point of (1), we have kk -2m(h+h')=0; which, by (3), becomes 9m+(h+h')=0, or (-m)-h=h'-(-m); which equation shews that the point of line 7'P, whose abscissa is (- m), is half-way between the points 7' and P; 7.e. TP is bisected by the directrix. 245. To find the equation of the parabola referred to the ‘foot of the directrix (#) as origin, and the two tangents drawn from H (which, as we have seen, are at right angles to each other) as co-ordinate axes. We make F the origin by putting x- m for z, in the equation yf = 4mz, and we make the two tangents the axes of co-ordinates by putting, x cos (- 45°) — y sin (— 45°), for 2, and, z sin (- 45°) +y cos(- 45°), for y ; hence the equation of the parabola referred to the new axes, is "Z-—y\ fa+y (Syar nen casna): or (a- y)'— 4mV2 (a + y) + 8m’ = 0; which, putting 2m V2 =a for brevity, may be reduced to the form (x+y) - 2a(x+y)t+a@ = 4zy, or L+y-aQ=+2 vVe2y, or Ve+vVy =Vva, which is the equation required in a remarkably simple form. 102 PLANE CO-ORDINATE GEOMETRY. 246. A parabola whose focus is given touches a given line, to determine the locus of its vertex. Let S (fig. 79) be the given focus, and BC the given line; draw SQ perpendicular to BC, and let A be the vertex of the parabola. Then, SQ being a perpendicular from the focus upon a tangent, Q must be a point of the line drawn through the ver- tex perpendicular to the axis; therefore SAQ is a right angle. Hence A is always on the circumference of the circle described on SQ as diameter; which circle is therefore the locus required. 247. If TP (fig. 80) be any tangent of a parabola, P’Q any other tangent meeting 7P in Q; then P’QS is an invariable angle, whatever be the position of P’Q. Draw AY'Y perpendicular to AS to meet the two tan- gents at Y’ and Y. Then SYP and SY'P are right angles; therefore SY'’YQ is inscribable in a circle, and therefore! Y'QS = Y' YS, which is an invariable ris whatever be the position of P’Q. Q.=.D. 248. If TQ, TQ’, and QQ’ be any three tangents of a para bola, the points 7, Q, Q', and S, lie in the circumference of the same circle. (fig. 81). | By the last problem SQQ' = SYY', and SQ'Q = SY'Y; and, since SYT and SY'T are right angles, SYY'+SY'Y= YTY; hence SQQ' + SQ’'Q = QTQ’, which shews that Q, Q,, 7, and S, lie in the circumference of the same circle. 249. 'To draw a normal from a given point (ad) toa ae The equation of the normal at the point (A/) is 2m (y-k)+k(a-h)=0; and, if this line pass through the point (ab), we have 2m (b—-kh)+k(a-h)=0, which, since A = Bh , becomes é 4m g ke + (8m? - 4ma)k — 8m*b=0........ (1) : i? | which equation determines £4; and then, since / = meee have wm 4 A also. it ‘E | PLANE CO-ORDINATE GEOMETRY. 103 _ Since (1) is a cubic equation, we may in general draw three different normals from a given point to a parabola. 250. To determine under what circumstances two of the three normals in the preceding problem coincide. In such a case (1) must have two equal roots, and therefore, by the theory of equations,* we have Sha OI tai Utes. ete erate 2) which, putting 4° = 4mh, gives 8h+2m-a=0. Also, in virtue of (2), (1) becomes e+ 4m’b=0; a—- 2m hence we have (4m'b)' = 4m ; which is the condition that must hold between a@ and 8, in order that two of the three normals drawn from (ad) may coincide. Hence every point of the curve whose equation is ys = p(x — 2m), where p = : (2). 3 \m possesses the property, that two of the three normals, drawn from any point of it, coincide. This curve is called the evolute of the parabola, and sometimes the semicubical parabola. Its figure is shewn in (fig. 82.) 251. To find the lengths of the two tangents drawn from a given point (Ak) to the parabola. * Let a and 6 be two roots of the equation etpertgertr=0.......--. (3); then we have a*+pa?+qa+r=0, and 0°+pb?+qb+r=0; and therefore, subtracting and dividing by a ~— 6, we find atab+h+p(atbj+q=0...... (4). Now, suppose 6 to approach a in magnitude; then (4) will approach the form 3a°+Qpa+g=0...-.-s5-e- (5); and therefore, when 6 becomes equal to a, the condition (5) must hold. Hence, if two roots of (3) be equal to a, we have a+ pa+qa+r=0, and 3a*+ 2pat+q = 0, 104 PLANE CO-ORDINATE GEOMETRY. By Art. 210, we have the equation y2 9 2 C08 0—k sin 0 obra — 4h _ ; sin’ @ sin’ 6 ; : and if the roots of this equation be equal, 7 is the length of a tangent drawn from (Aé) to the parabola. Supposing then that this equation is a perfect square, we have _ 2m cos 0 - k sin 0 2 k= 4mh Rel ian = ep hae sin? 0 iar eri ah r =A (2) Hence, if we put k° - 4mh = c’, (2) gives sin 0 = £ Aopen r r and therefore (1) x sin’ @ becomes, by (2), c = 2m y(r’ - c’) — ke, \2 and ». r=’ (° i =) + ut, |\ 2m which is the expression for the squares of the two required lengths, the value of c being + V(k* — 4mh). 252. If Tbe the point of intersection of two tangents drawn at the points P and P’ of the parabola; to shew that the distance of 7’ from the axis is an arithmetic mean between the distanecs of P and P’ from the axis. Let Tbe (hk) and P, or P’, (xy), then | ky = 2m (x + h), which, since y* = 4ma, becomes y — 2ky + 4mh = 0. Let y and y’ be the two roots of this, then y ty = 2k, which shews that & is an arithmetic mean between y and ya rae Of Oe CHAPTER IX. OF THE ELLIPSE. PROPERTIES OF THE ELLIPSE NOT CONNECTED WITH TANGENTS OR DIAMETERS. OF THE ECCENTRIC ANGLE. PRO- PERTIES CONNECTED WITH TANGENTS. PROPERTIES CONNECTED WITH DIAMETERS. VARIOUS PROBLEMS. 253. The equation of the ellipse referred to its centre (C’) as origin is, by Chap. v1., 4? y? Tal tees a being the semi-axis major (C/A), and 6 the semi-axis minor (CB). The polar equation referred to the focus (S) is, by Article 177, y ss bccieicoss Mic =, 1 being the semi-latus-rectum, and e the eccentricity. We have also, by Art. 176, 2 NG@- ae, EC=- , P=aa@(1-e), l=a(ll-e - , e The curve is perfectly symmetrical with respect to 44’ and |BB’, so that there is another focus S’ and another directrix | E'K' (fig. 57), corresponding to S and EK, CS’ and CE’ being respectively ae and eS e We shall now, by means of these equations and formule, in- vestigate the various remarkable properties of the ellipse. Properties of the Ellipse not connected with Tangents or Diameters. Prop. LXIII. 254, 'Lo find the distances of any point of an ellipse from the two foci. — 106 PLANE CO-ORDINATE GEOMETRY. Let (zy) be any point (P) of the ellipse ; then, since ae and 0 are the co-ordinates of S’, we have S"P? = (@ —- aey + y’, 9 ne which, since y* = 0° (1 - a) (a° -z°) (1 - e&), becomes SP? = ex’ — 2aex + a =(a - exy’, therefore S'’P =+(a- ez). S'P here means the absolute distance of P from S' without regard to sign; hence, since e is <1 and w 2 2 In the equation = a or = 1, put z =r cos 0, y=7 sin 0, and it becomes ) , f/cos’@ sin’? r ( eben As Te ey ates toes (1), ab or ”~ TG cos 04 a sin’ 8) Ge O1e)e sie fee a (2), which is the equation required. It is sometimes given in the | form ii v1 - é) ~ V(1 = & cos? 6)" 267. Cor. By (1) we have Ly 9 COB Oey isin ag Poe Sar Aba and if 7 be the radius vector drawn at right angles to r, we have, putting 0 + 3 for 0, 1 sin’ 9 cos’ 0 Eula aes, Hence, by addition, we find 1 1 a 1 1 Yr yay iat b? oe ae PLANE CO-ORDINATE GEOMETRY. lid It appears therefore, that the sum of the reciprocal squares of two radii vectores, drawn from the centre of an ellipse at right angles to each other, is invariable. Of the Eccentric Angle in the Ellipse. Prop. LXX. 268. The co-ordinates x and y of any point of an ellipse may be put in the form z= a@ cos 9, y= b sin ¢. x Let us assume, as we may, that — = cos p; ve. let @ be the a . e L . ° . e angle gv hose cosine is = ; then, substituting in the equation 2 x’ a & at gnh 2 we find i = 1 —- cos’ ¢ = sin’ 9, and therefore = sin 9. Hence we may assume z = a cos ¢, and y = 6 sin ¢. 269. Cor. 1. If we describe a circle AP'A’ (fig. 85) on AA’ as diameter, produce the ordinate MP to meet it in P’, ey Rca Ne and join P’ and C; P’CM is the angle whose cosine is = Boece. CM = Hence, if we produce the ordinate to meet a circle described on the axis major as diameter, and draw a line from the point of meeting to the centre, ¢ is the angle which that line makes with the axis major. 270. Cor. 2. Draw PQR parallel to P’C to meet CB’ in R and OA in Q; then PQ= P’'C=a, and PQA' = P'CA' = 9. Hence ¢ is the angle which a line, equal to a, drawn from P to BB’, makes with AA’. Since y = 6 sin ¢, we have 6sin¢=PM= PQsin PQA’ = PQ sin 9, therefore Path O, 1412 PLANE CO-ORDINATE GEOMETRY. Hence @ is the angle which a line, equal to b, drawn fror P to AA’, makes with dh Hence, if a line be drawn from any point P of an ellipse to BB, equal 1 in length to a, the part of it intercepted between P and AA’ is equal to 0. . 271. Cor. 3. Hence, if 4A’, BB’ be two lines at nght angles to each other (fig. 86), and PQ# any line intersecting et so that PR = a, PQ= 4; then P is a point of the ellipse whose axes coincide with 4A’ and BB’, and are equal to 2a and 26. 272. Cor. 4. Hence, if we suppose PQR to be a ruler having a pin at R and another at Q, and if AA’, BB’ be grooves in which these pins run, a pencil at the point P will trace an ellipse. In this manner an instrument for drawing’ hips called an elliptic compass, has been constructed. 273. The angle @ we have here introduced is of con- siderable importance, &s will appear, in proving several pro- perties of the ellipse. It is made use of in Astronomy, and has been termed the eccentric anomaly, certain angles being called anomalies by astronomers. Instead of the word anomaly we shall employ the word angle, and accordingly call ¢ the eccentr 0 angle of the point P. If therefore ¢ be the eccentric angle of any point of an ellipse, a cos ¢ and 6 sin ¢ are the co-ordinates of that point. Properties of the Ellipse connected with Tangenits. Prop. LXXI. 274. ‘To find the angle which the line touching an clips at any point (h/) makes with the axis of z. As in Art. 135, if we suppose Q and Q' in Art. 268, to coin- cide, the line PQQ!’ becomes a tangent at the point Q; and : then have, by equation (2), ; en ee ae i > $6 Ff PLANE CO-ORDINATE GEOMETRY. 113 which, if we suppose P to coincide with Q, gives V = 0, or h cos 0 k sin 0 . “ ee ) 2 and therefore far Gi” f +) ak Hence — yh. is the tangent of the angle which the line touch- ng the ellipse at the point (24) makes with the axis of z. 275. Cor. 1. Hence the equation of the tangent at (hk) is, ny Art. 60, h i | = a @-h)t+ a y-h=0, ( ech 6; ene he which, since at ibaa 1, becomes he ky eC ous Be a 1 ° C1628 616. 6. 6 8-86 ® ° Gn) : * This result may also be obtained, as in the case of the parabola, as pllows. _ The equation of the line joining any two points (hh), (A’k') of the ellipse is | ki-k y—h=a(@-h)e esse, (4). Tow, subtracting the equations h? I? h/2 }:2 1 pan sia Oye pe Ki? — ie h® — jf? _K-k hth Per Ca MOM EHS az ad therefore (4) becomes ee ee eh (any; a Ry erage a hich equation may be made to differ from ~he-5 2 (@-h) (5) y = aA ji oo smell sys: s. 6 ) little as we please, by bringing the point (A/A’) sufficiently near to (hh). sence, reasoning exactly as in the case of the parabola, (5) must be the ngent at (hk). (5) may be reduced, as in the text, to the form 114 PLANE CO-ORDINATE GEOMETRY. 276. Cor. 2. Hence, if 7’ and 7” be the points where the tangent at (hk) meets the major and minor axes, we have, by Art. 57, a AES. it CT = va Or = ae Prop. LX XII. o77. If (ax + By =~) be the equation of any tangent to the ellipse, to determine what relation must hold between a, (3, and y. Let (hk) be the point of contact; then the two equations Loam SOBs) Bee ds ili EA aay Ces must represent the same line. ‘Therefore a ee e k — = 3 BR? H and therefore, since (Oy a a (elias tenia ay. a Hence the required relation between a, (, and y is y =@a’ + B".+ \ = 1, we have * This result may be proved geometrically as follows. ) On 44’ (fig. 87) as diameter describe a circle, produce any two oré i-| nates, MP and M'P’, to meet it in Q and Q’, and draw the lines Q'Q, PF D then these lines produced must meet CX in the same point 7; as is eviden n from the fact, that IP: MQ=M'P': M'Q, by Art. 261. Now if P and Q' be supposed to approach P and Q, PT and QT will ultimately ! become tangents at P and Q. But, if Q7 be a tangent at Q, COT is ¢ a right aH and therefore O7': CQ=CQ: CM; which, since CQ=a, gives CT= “ . And in the same way we may shew that C7" = & + This result may be obtained independently in the following manner. — ! Let (zy) be either of the points of intersection of the line and ellipse represented by 2 ay ar +Py=7..... (3), settee nee Be (4). Then (3) — (4) 7’ gives (o-2) 2 + Basar + (6 -3) y =O... tam PLANE CO-ORDINATE GEOMETRY. 115 278. Ify=az + be the equation of a tangent, this relation gives y’ = aa’ + 6°: hence the equation of any tangent of an meee may be put in the form y= ax + V(a'a’ + 0°). The double sign shews that two tangents may be drawn at the same angle to it axis of z; which is manifestly true. Prop. LX XIII. 279. To find the equation of the normal at any point (A/) of an ellipse, and the portions it cuts off from the axes. 2 b hi . . ; ) with the axis of z, ak The tangent makes an angle tan” (= and the normal is the perpendicular to it drawn from the point (hk); therefore the equation of the normal is ak (x-h)- Uh (y-k)=0, or Ca yad 8, which is the equation required. Hence, if G and G" be the points where the normal meets the axis major and the axis minor, we have 2 2 ro] ie iat LS yop BY Fae Oe Cie tat a aaa CG RP k a Obs which are the portions required. 280. Cor. 1. Hence the normal at P bisects the angle SPS". For we have (in fig. 88) SG = SC + CG = ae + ¢h = eSP, by Art. 254, SG = S'C —- CG = ae - &h= eS'P; - This equation gives, in general, two values of f but, if (3) be a tangent, the two values ought to become equal, and therefore (5) ought to be a perfect square ; which gives 2 Y’ 2 Y 222 2 2.2 id ee b * oto e- = =a*6*, or y'=a'a’ + 08", as above. 12 , 116 PLANE CO-ORDINATE GEOMETRY. therefore SG. S'G=SP28Lf, which shews that PG bisects the angle SPS’. Q.E.D. * 281. Cor. 2. Hence the tangent at P makes equal angles with SP and S’P. This result may be proved geometrically by means of the property, SP + S'P = 2a, in exactly the same way as in the case of the parabola. See Art. 217, note. f Prov. LXXIV. . 282. To find the locus of the intersection of a tangent with a perpendicular let fall upon it from either focus. By Art. 277 the equation of the tangent is az + By = + V(a@’a’ + OP") . 2 Jaca CL and the equation of a line drawn perpendicular to this from S or S', is B(x + ae) - ay = 0, or B2—ay=i Bd - 0’)... eee (2): (1)? + (2) gives immediately, dividing out a’ + fp’, Dae iy jie Mo Afi tie ee a Sp * The following is a more analytical proof of this property. Let 4 and 6' be the angles which SP and S’P make with 44’; then we have k: k tan 6 = oo ; h+ae’ fee h—-ae’ : 7. " and therefore tan (@+ 6’) = ge ay : —@ te k? : which, if we put h=acos #, and therefore k=6 sin + (Art. 268), becomes 2ab sin ’ cos — ea Game) cos? ¢ -a®+6°- 6? sin? > Shaun swise _ __2ab sin ¢ cos ~ 8? cos? ¢ — a sin? >” Now, if ¥ be the angle which PG makes with 44’, we have ah _ asin > bk bcos?’ 2ab sin ? cos > b cos? 6 - a sin? ¢- tan yy = and therefore tan 2 Wy = e . R Hence 6 + #’ = 2¥; which shews that PG bisects the angle SPS’. WEA Rie PRL NS ELE hae te Li PLANE CO-ORDINATE GEOMETRY. 117 Hence the locus required is a circle described on the axis major as diameter.* ’ 283. Cor. This proposition suggests the following method of drawing (geometrically ) a tangent, or rather two tangents, to an ellipse from any given point U (fig. 90). Join S and U, and on SU, as diameter, describe a circle cutting the circle described on AA’, as diameter, at Rand FR; then the lines drawn from U through R and &' will touch the ellipse. This is evident, since SRU and SR'U are right angles, and therefore, by the proposition just proved, the lines drawn from U through #& and F' must be tangents to the ellipse. | Prop. LXXV. 284. ‘l'o determine analytically the tangents of an ellipse which pass through any given point. The equation of the tangent is y—ax=+ Va’ + 0’). Suppose it to pass through a given point (AA), thén we have (& — ahy = aa’ + 8’, - 2hk b’ — }? or rip el peceiemhl, ty yt 93 Saale Genk 7 ea which equation gives two values of a; let them be a and a’; then the equations of the tangents required are Me kj=al(e—h)....(2), “y-k=a (e- hk). ...(3): 285. Cor. To determine the locus of the point of intersec- ion of two tangents at right angles to each other. a Pe ae cea ehY: * This may be shewn geometrically as follows : Let SY be the perpendicular from S upon the tangent (YP) at P fig. 89); produce S'P and SY to meet in Q. Then, since QPY = SPY, dy Art. 281, we have PQ = PS, and therefore, by Art. 255, S’Q = 2a; and, since SC = CS' and SY = YQ, we have CY =3 S’Q=a. Therefore ve is 1 point on the circumference of the circle described on the major axis as liameter. 118 PLANE CO-ORDINATE GEOMETRY. If (2) and (8) be at right angles, we have aa +1=0; yk 1 iieeh servis but, by ( oe aa a sae h? ’ therefore by 7 “a 41 =00- es eB) or BP+ikR=aa +06’. Hence the locus of the point (24) is a circle described with C as centre and v(a@’ + 6”) as radius. This may also be shewn very readily as follows. The equation of any tangent 1s | ax + By =+ V(a'a’ + B20")... eee eee (1), 9 and the equation of a tangent at right angles to it 1s Bx — ay = + VaR? + Ba’)... cece eeeee (2), 9 (1Y + (2) gives eS) e+y=a+ b’, which is the same result as before. ? Prop. LX XVI. 286. If p and p’ be the pyrpandicness from S and S' up@ any tangent, to shew that pp’ = 6°. The equation of the tangent being ax + By = y =V@a’ + UP"), we have, by Art. 69, the points S and S’ being (- ae, 0) anc (ae, 0), Yt aae uy = aes. : nee a V(a° Va? + 8°)” therefore pp = 1S ih) a+ 3° 073 4 Ba _} a’ + oy 987. Cor. If we denote SP and S'P by r and 1, iti evident, since SP and S'P make equal angles with the tanga at P, that po? Q. E. D. | :, PLANE CO-ORDINATE GEOMETRY. 119 multiplying this by pp’ = 4°, we find Y 2 2 p= fei ihe r ° ’ = Sata since 7 +7 = 2a, which is an expression for the perpendicular from S upon the tangent at P, in terms of SP. Prop. LX XVII. 288. To find the perpendicular from C upon the tangent, in terms of the angle which the tangent makes with the axis of 2. The equation of a line making an angle 6 with the axis of z and at a perpendicular distance p from the origin is, by Art. 57, y cos 9- x sin 0=p, and, by Art. 277, if this be a tangent, we have p= a sin’ 0 + 8° cos’ O, which is therefore the expression for the perpendicular required. It is often put in the form p=@(1 - & cos’ 6), by substituting for 0” its value a’ — ae’. Prov. LX XVIII. 289. To determine the equation of the line joining the | points of contact of the two tangents drawn from a given point | to an ellipse. Let (hk) be the given point, and (zy) either of the points of contact ; then the equation of the tangent gives hua ky ied ide and, reasoning just as in Art. 139, we may shew that this is the _ equation of the line joining the two points of contact. 990. Cor. 1. Hence, reasoning exactly as in Art. 140, we _ may shew that the locus of the intersection of the two tangents, ‘drawn at the extremities of any chord passing through (A4), is ha | hy _ 5 a Lb 120 PLANE CO-ORDINATE GEOMETRY. 291. Cor. 2. Hence the locus of the intersection of the two tangents, drawn at the extremities of any chord passing through S, is the directrix HH. For, putting 4 =0, h=— ae, the equa- tion of the locus becomes z = - = , which represents HA. Properties of the Ellipse connected with Diameters. Prop. LX XIX. 292. ‘To determine the diameter of a given system of par alle ; chords in an ellipse. . Let 0 be the angle which any chord makes with the axis of z, and (zy) its middle point ; then, by Art. 264, we have . z cos 8 _ ysin kee B =O) eee 65) ' a and consequently, if we suppose the chord to move parallel to itself, and therefore 9 to remain invariable, this equation repre-_ sents the locus of its middle point. Hence (1) is the equation of the diameter of the system of parallel chords which make an | angle @ with the axis of z. 293. Cor. 1. If we suppose the chord to move parallel to_ itself until its extremities coincide, it then becomes a tangent 4 and hence it follows, that the tangent at the extremity of any diameter is parallel to the chords of that diameter. This may also be easily seen from the equations of the diameter and tan- gent ; for, if (Ak) be the extremity of any diameter, and 6 the angle its chords make with the axis of z, we have, by the equa- J tion of the diameter just obtained, Bh | ah’ hence, by Art. 274, @ is also the angle which the tangent at (hk) makes with the axis of x. Therefore the tangent is parallel to the chords. #94. Cor. 2. The equation (1) shews that every diameter | of the ellipse passes through the centre (C’‘). Therefore every. chord drawn through C is bisected at C. tan @ = — PLANE CO-ORDINATE GEOMETRY. 12} 295. Cor. 3. If 0 be the angle which the diameter (1) makes with the axis of z, we have 6° cos 0 a and therefore tan @ tan 6’ = — us tan 9’ = - ~ — a sin 0’ a Hence, if @ and @’ be the angles which a system of parallel chords and their diameter respectively make with the axis of z, we have the relation 2 er Ota Ge ete Cay a 2 from which we may find either 0’ in terms of 0, or 8 in terms of 0. PROPS LX AX. 296. If one diameter be parallel to the chords of another, the latter diameter will also be parallel to the chords of the former. Let us denote the two diameters by D and D,, let 0, 0, be the angles which their chords, and 6’, 0,' the angles which they themselves, respectively, make with the axis of z. ‘Then, by (2), we have tan @ tan 0’ = tan 6, tan 0,’, which shews that, if tan @ = tan 0/, then tan 0, = tan 0’. Hence, if the chords of D be parallel to D,, the chords of D, will also be parallel to D. Q.5.D. Two diameters thus related are called conjugate diameters. 297. Cor. 1. Hence, if @ and @’ be the angles which two ‘conjugate diameters make with the axis major, we have Ae tan 9 tan #' =- —. a 298. Cor. 2. If (ay) and (2'y') be any points on the two ‘conjugate diameters respectively, and therefore z =a falta / MS = tan 0’; we have yy b° ce yy’ “f= — or’-— +57 =0. 2x a a i? 123 PLANE CO-ORDINATE GEOMETRY. Prop. LXXXI. - 299. If @ and ¢’ be the eccentric angles of the extremities : two conjugate diameters, then ¢’ = ¢ + 90°. Let (zy) and (z'y') be the extremities P and D of wil conjugate diameters CP and CD (fig. 91), and » and o the corresponding eccentric angles, then J xZ= a COs 9, y=bsin 9, ‘=a cos @’'; y= being, and therefore, by the preceding Article, we have cos ¢ cos ¢' + sing sing’ =0, or cos(p ~ p)= 9, which shews that ¢'- @ = 90°. Q.E.D.* This property of conjugate diameters, with reference to the eccentric angle, will be found of great use in all problems relating to conjugate diameters. ; 300. Cor. Hence i E: x a cos (# +5) and y =6 sin +2) b cos ¢ = ae : Se p 2 g= a ’ which formule determine the co-ordinates of D in terms of those of P. : tee es lI -asing=- 5% Prop. LX XXII. 301. The sum of the squares of CP and CD, and the area of the parallelogram completed upon CP and CD, are invariable. Let CP=r, CD=7r', PCA=0, DCA=9, (fig. 91). Them r=xt+y =a cos ¢+ 6’ sin’ 9, | and, putting ? + 2 for ¢, we change 7 into 7; therefore ry? = a’ sin’ p + 8° cos’ . Hence OM SOM ee Se AF ut (1). * Hence the following simple construction for determining two conjugate diameters of an ellipse. | On 4.4’ (fig. 92) as diameter describe a circle 4'P'D'A, from C draw any two lines CP’, CD’ at right angles to each other, and draw the or ding P'PM, D'DN; then CP and CD are conjugate diameters. PLANE CO-ORDINATE GEOMETRY. 123 Again, if A be the area of the parallelogram completed upon CP and CD, we have A =rr' sin (0 — 0) = 77’ (sin 6 cos 8 — cos 6’ sin 8) pas xy’ al xy = ab sin(@' — ¢). Hence, since ¢' — 9 = 3 , we have J CLS eH ie Oe (2) (1) shews that the sum of the squares of CP and CD is invariable, and (2) shews that the area of the parallelogram com- pleted upon CP and CD is invariable. Q. E.D. 302. Cor. 1. Since the tangents at P and D are respectively ‘parallel to CD and CP, it follows that the area of every paral- lelogram circumscribing an ellipse, having its sides parallel to two conjugate diameters, is the same. 303. Cor. 2. In the above expressions for 7” and r’, putting 1 -—cos’ ¢ for sin’ ¢, we have r=)? + ae cos’ ¢ = 8 + €2’, r= a? — ae cos’ @ = @ — &2’. Hence, since, by Art. 254, SP=a+ezx, S'P=a- ex, we have SP.S'P =r”, or BS eC Prov. LX XXIII. 304. To find the equation of the ellipse referred to any two ‘conjugate diameters, CP and CD, as axes of co-ordinates (fig. 93). Draw any chord QMQ' parallel to DCD’; assume OM MQ=y,CP=a,CD=8'. Then, by Art. 265, we have MP.MP MQ.MQ CPVOP < CD.CD4 But CP = CP’ =a, MP=a'-2, MP'=a' +2, CD=CD =6, and MQ = MQ' =; therefore (fhe eRe i y 2 12 5” 3 124 PLANE CO-ORDINATE GEOMETRY. 2 2 x or Set fit cece eeeee ee eeees OD) which is the equation required.* | Hence it appears that the equation of the ellipse referred 1 to 4 any two conjugate diameters is of exactly the same form as that referred to CA and CB. Prop. LX XXIV. | 305. The ellipse being referred to two conjugate diameters a to find the equation of ie line touching it at any proposed point (A). a Proceeding exactly as in the notes to Arts. 214 and 275, we may shew that the equation of the tangent required is el ol AS * This result may be obtained by transformation of co-ordinates in the — following manner. Tf, in the equation = ~~ hae re = 1 (which is referred to CA and CB), we put x cos 84+y cos 6 for a, oe x sin @+y sin 6’ for y, it becomes Pa (x cos 0+ y cos oa (x sin @+y sin Oro a Bae RE Lek 42 Me ede bs = and this substitution amounts to turning the axes of co-ordinates round the origin, until the new axes of x and y make angles @ and 6’ with the old axis 4 of x. (See Art. 110.) ; Now, 9 and © being disposable, we may assume that cos 8 cos 6’ sin @ sin 0 2 ai ah) ae (which amounts to supposing the new axes to be conjugate dianieters) ; and then (2) becomes cos’ 6 sin? 6 cos? 6’ sin? 6’\ —7"7 ( a ( ae tgs \y=1...@) = A; a b? But, by Art. 266, we have 3 cos? 0 sin? 6 es con’ 0’ ain? 6 1 a pA MT aT AO A (rs 2 hence (3) becomes <+ Yop if iq pion coincides with the equation in the text, observing that =a’ and = 0, ! $f PLANE CO-ORDINATE GEOMETRY. 125 a 306. Cor. 1. Hence, as before, the portions cut off by the tangent from the conjugate axes are a’ 6” — and —., h k _ 807. Cor. 2. Hence, also, the equation of the line joining the points of contact of the two tangents, drawn from any point (Ak), is he hy 308. Cor. 3. Hence we may shew, exactly as in the case of the parabola (see Arts. 236, 237), that the line joining the points of contact of two tangents, drawn from any point of a line parallel to CD, always crosses CP at a given point. Also, that, if two tangents be drawn from any point of a diameter produced, the line joining the points of contact is an ordinate to that diameter. Various Problems connected with the Ellipse. 309. An ellipse being traced upon paper, to determine its centre. Draw any two parallel chords PP’, QQ' (fig. 94), bisect them in Mand N, and draw DD’ through M and N; then DD’ is a diameter, and therefore its middle point (C) is the centre required. 310. An ellipse and its centre being given, to find its axes. With the given centre C (fig. 95) as centre, describe a circle cutting the ellipse in P and P’, join PP’, draw ACA’ and BCB perpendicular and parallel to PP’; then AA’ and BB are the axes required. 311. To find the locus of the middle points of all chords passing through a given point (AA). If (zy) be the middle point of the chord which makes an angle @ with the axis of z, we have zcos@. ysin@ , Si aE B ive dtc. eas a 126 PLANE CO-ORDINATE GEOMETRY. and if (zy) be a point on the line drawn through (AA) making an angle @ with the axis of z, we have (7 —-h)sin 0-(y~—k)cos0=0........(2)m Eliminating @ between (1) and (2), we find 3 r(x —h = ae 1 ed aa a i al 2h cs Bi vonk oe Solas | cAbie which is the equation of the locus required. It represents an ellipse whose centre is (5 ; 5) , having its axes parallel to those : of the given ellipse, and equal to 0B BY 8 RB) 2 / He Ne a hay irae) \/ ieeatiee BLO 1f/6 = PSSe GueaP Ss tolehewithal Pele Fey = ieee 1l+e ‘M We have 608 Ole oe at SP a+ex , 0.1 cos 6) ae eg ae ix 2 1+cose0 atexriaciz Q@Q+ze 1+¢6e ° ° ae — 63 Similarly cos 0 = a— ex Oo. Sut ere Andes Satan se Hence tan : tan PLANE CO-ORDINATE GEOMETRY. 127 313. If 6 and 6’ be the angles which two conjugate semi- diameters 7 and 7’ make with the axis major, to shew that sin(@'+0) r-r Bin (One 0 Vi tase De 7 rr sin (0+ 0) zy + xy rr sin (0-0) zy — ay ab (cos @ sin ¢' + cos @ sin #) ab (cos @ sin ¢' — cos ¢’' sin ¢) (see Art. 300) _ sin (¢' + @) sin (¢' — ¢) = cos 2%, since ¢'=o+ 5 Also 7 =a’ cos’ $+8' sin’¢, 7° =a’ sin’ ¢ + 0’ cos’ 9; therefore y” — 7” = (a’ — b’) cos 26. sin(0+0) _ r- 9” sin (0 -— 0) a’? — 6. Hence 314. The circle described on S’P as diameter touches the circle described on 4A’ as diameter. Let (hk) be P; then the equation of the circle described on | S'P as diameter, is ( ae + 2) ( 5) (- - “y (5) xL- +{y-—)={——)+4+ [=], 2 2 2 2 or 2 —(h+ae)a+y’—ky+aeh=0.......4. (1). Also the equation of the circle described on AA’ as diameter is (AYE 108 eC ag eA SE OCMRE DRE Cy. At the points of intersection of (1) and (2) we have, subtract- ing (1) from (2), (h+ae)a+hy=@+aeh.......... (3), which is therefore the equation of the line drawn through the points of intersection. Now, if p be the perpendicular from C upon (3), we have (@+achy¥ a(SP) _, . a de ithuey aie a USP) 128 PLANE CO-ORDINATE GEOMETRY. Hence (8) is a tangent of (2), which cannot be unless the points of intersection of (1) and (2) coincide. Therefore (1) and (2) touch each other. 315. If S'P =r, PS'A = 9; to find r and @ in terms of the eccentric angle () of the point P. If z be the abscissa of P, we have r=a—ex=a(l1—€Cos d)........ Cj oe eee t which are the values of 7 and @ required. These values are of considerable use in Astronomy; (2) being generally expressed , in the form 0 (l+e \ | tan —=,| > Vie) 316. If PQR (fig. 96) be drawn so that PQ=6, PR= a | and if the rectangle CU be completed ; then PU is the normal alee | x- ae cos @ - e 7 1—ecos¢ | cos = PM+ QU. CM — CQ,’ which, if PQA = ¢, becomes, since PQ = 6, PR =a, bsing+(a-—b)sng a tan ¥ = J cosg —(a—b)eosg B® tan yp = Now, if yw’ be the angle which the normal at P makes with CA, we have . Hence y'=y. Q.8.D. 317. If the axes of an ellipse be in the proportion of 1:2, 3 any parabola described on the axis minor as axis, and having its | vertex at the centre, will cut the ellipse at right angles. | Let the equations of the ellipse and parabola be a 2 ye — 4+. = 4mx a } y >) a PLANE CO-ORDINATE GEOMETRY. 129 and let @ and ¢ be the angles which tangents, drawn to the two curves at their point of intersection, make with the axis of z. Then we have bx 2m tan@0=-—, tang= —, : y 2mb* and .. tan 9 tan ¢ = - het ay 2 OTA a? since y = 4mz. Hence, if 6: a::V/2:1, we have tan 9 tan ¢ = - 1, and therefore the two tangents are at right angles, or, in other words, the curves cut each other at right angles. 318. To find the locus of the middle point of a chord of the curve Az’ + By’ = C, the length of the chord being given. Proceeding just as in Art. 262, we find 7” (A cos’9+B sin’ 9) + 2(Ahcos 6+ Bksin 6) r+ Ah’+Bh’-C=0 ; and, if we suppose (h/) to be the middle point of the chord 27, we have Al COS G+ DA S10. f = Oe cloe er Ch) and therefore y’(A cos’? 0+ B sin’ 0) + Ah’ + BRK’ - C=0....(2). ‘1) gives tan 0 = - alt , and then (2) becomes ee Bk 2 ABR + BAN Ah’ + BR’ Now if we suppose r to be invariable (equal to ¢ suppose), and if we put z and y in place of A and 4, this equation becomes (Az? + By’ - C)(4’c’ + By’ - CAB) = CABC...(8), which is the equation required. + Ah’ + Bhe- C= 0. 319. If we suppose (= 0, A=-a’ and B=1, the given 2quation becomes (y — az) (y + az) = 0, and (3) becomes (y? — a?x”) (a’x* +. y? — a’c*) = 0. Hence if lines of a given length (2c) be drawn between two ines (y = az) and (y = — az), the locus of their middle points is | y wep Ottands. act y? be ae, K 130 PLANE CO-ORDINATE GEOMETRY. the former equation representing the two lines themselves, a C 4 the latter an ellipse whose axes are — and ac. The former a } equation represents the locus of the middle points of those chords which have both their extremities on one of the lines; and the latter of those which have one extremity on one line and the other on the other. ; 320. CX (fig. 97) is a fixed line, C a fixed point upon it, CQ, QR two equal lines of a constant length, and P a fixed poi nt on QR: to find the locus of P. P| Let CM (=x), MP (= y) be the co-ordinates of P, let QCR = 0, CQ+ QP =a, and PR= 4; then we have | z= CQ cos 0+ QP cos 0=acos 6, andy =46sin 8. Hence tone a The locus required is therefore an ellipse, whose axes are CQ + QP and PR. 4 Elliptic compasses might be made upon the principle sug gested by this problem. : CHAPTER X. f THE HYPERBOLA. PROPERTIES OF THE HYPERBOLA CORRESPOND- ING TO THOSE OF THE ELLIPSE. ASYMPTOTES. PROBLEMS. 821. The equation of the hyperbola referred to its centre 7) as origin is, by Chap. vi., ; 2 2 being the possible axis (4A’). 6 does not now represent the ortion the curve cuts off from the axis of y, since that axis does %t meet the curve. _ The polar equation referred to the focus (S’) is 2 y 1—ecos 0’ deing the semi-latus-rectum, and e the eccentricity. ‘We have also, by Art. 176, Y a 2 2/2 st OF 7= — ae, Cas b’ = a’ (e’ — 1), PTA Neh Nor se : The curve is perfectly symmetrical with respect to the co- ‘dinate axes, so that there is another focus S’ and another rectrix E'K’ (fig. 59) corresponding to Sand EX. | We shall now, by means of these equations and formule, vestigate the various remarkable properties of the hyperbola, amy of which correspond so exactly to those of the ellipse, that will be sufficient merely to state them, and refer to what has en already done in the case of the ellipse in proof of them. 132 PLANE CO-ORDINATE GEOMETRY. Properties of the Hyperbola not connected with Tangents or Diameters. 2 | Prop. LXXXV. : 399. To find the distances of any point of a hyperbola from the two foci. | Let (zy) be any point (P) of the hyperbola; then, since 7 and 0 are the co-ordinates of S, we have SP? = (x - ae) +y’; 2 which, since y’ = 0 (S - 1) = (e — 1) (2 - a’), becomes ¢ SP? = &x — 2aex + @ =(a - exy, ] therefore SP =+(a- ex); SP here means the absolute distance of P from S'; hence, sine eis>1, anda >a, and therefore (a- er) a negative quantity we must reject the upper sign. We have therefore In like manner we find, since — ae and 0 are the co-or ila of S', S'P=(x+aeyr+y’, , = ¢a + 2aex+a’, and therefore S’P = CL + GDirveseees . (2)34 (1) and (2) are the expressions for the required distances. 323. Cor. By subtracting (1) from (2) we find immedialfl S'P — SP = 2a. | * (1) and (2) may be deduced geometrically, as follows. In fig. 98, drawing QPQ’ parallel to AA’, we have | SP = eQP =e(CM - CE) | =ex-—a, since CH= -. Also S’P =eQP =e (£'C+ CM) = ex + a. We may also prove geometrically, that S’P-— SP =2a, as follows, S'P ~- SP =e (QP - QP) =e (£'C+ EC) = 2a. : PLANE CO-ORDINATE GEOMETRY. 133 Hence, in the hyperbola, the difference of the distances of my point from Sand S’ is always equal to the axis major.* We may obtain the values of SP and S'P very easily from Art. 204, as in the case of the ellipse. See Art. 258. Prop. LXXXVI. 324. If PSP’ be any chord drawn through S, the semi-latus- ectum is a harmonic mean between SP and SP’. This may be proved exactly as in Art. 207. Prop. LX XXVII. 325. Two given lines being drawn through the vertex A, ‘o find the equation of the line joining the other two points in which they intersect the hyperbola. Let the equations of the two lines be y=a(t-a@), y=a (t-@). Then, as in Art. 260, the equation required is ° 4 pet Dat a) y Paa (2a )= 0. We may deduce from this equation the same conclusion as n Art. 260. Prov. LX XXVIII. 326. To determine the length (7) of a line drawn, at an ingle @ to the axis of xz, from any point (h/) to the hyperbola. As in Art. 262, we find Cire te 2 Vi ricte 4 OG eee cc sare oak Ly ere Ue cos’ @ _ sin’ 0 va hcos@ sin 8 a a” 6° : z a 6b? > se ie =+~-5-1 Uo nee * Conversely. To find the locus of a point P (fig. 99), the difference of vhose distances from two fixed points S and S" is invariable. Proceeding as in the Note to Art. 255, we have r’—7 = 2a, and therefore r” = 7? 4 4a? + 4ar, also r” =r? + 4c? + 4cr cos 8, (ESA Cy, nd therefore, putting c= ae, we find ALR Creed Yad ~ 1-e cos 0’ shich, since ¢ is evidently > a, and therefore e> 1, represents a hyperbola, 134 PLANE CO-ORDINATE GEOMETRY. Also PO. PQ = 2 (2), ug 5 PO. Po eee Also the condition that (2é) con be the middle point of the chord, which makes an angle @ with the axis of z, is h cos 6 k sin 9 2 ec 3 oO A a b° f Prop. LX X XIX. j 327. If QQ’, RR’ be two chords of a hyperbola, and P thei point of intersection, the ratio PQ.PQ’': PR.PRH is not altered by moving each chord parallel to itself, and so ae the position of P in any manner. The proof of this is exactly the same as in the case of the ellipse. See Art. 265. Prop. XC. 328. If QQ', RR, Q,Q, be chords of the hyperboll 2 2 2 : oat 1), and RR, a chord of the hyperbola (-2+5- } Q,Q,, RR, being respectively parallel to QQ, RF’, P being the point of intersection of QQ', RR, and P, that of Q,Q,. Fee ne Gu PO MEO) Bae GAMETe) & PR PR ~~ PR.PR, ; For, as in Art. 265, we have PQ.PQ & cos’ ¢-a sin’ o. : PR PR 6 cos’ 0 - a’ sin’ 0” % and, in exactly the same way, we find ; P,Q,. P,Q, _ - 8 cos’ ¢ +a’ sin’ % PR,. PR, & cos’ 0-a' sin’ GO 4 PQ.PQ__ P,Q. P,Q & Hence A CSS ENE aig Se eee ee eet ‘ § The latter trerte tate here Abrttbcteat 1s “galeede called the Conjugate of the former hyperbola, for a reason we shall pre: sently explain. By Art. 180, the conjugate hyperbola has the axis of y for its possible axis, as is represented in fig. 100, where B,, B, are the vertices of the conjugate hyperbola, and B,B,= 2b 4 “See Art. 180, Equation 3. PLANE CO-ORDINATE GEOMETRY. 185 Prorv. XCI. 829. To find the equation of the hyperbola referred to the centre as pole. As in Art. 266, we find the required equation to be isan Capewen et V(b’ cos’ 9 — a’ sin’ 0) T= Of the Eccentric Angle in the Hyperbola. (See Art. 268.) Prov. XCII. 330. The co-ordinates x and y of any point of a hyperbola may be put in the form | i= @ COS d, y =h v(- 1)sin ¢. , ; x As in the case of the ellipse, we may assume ~ = cos 9; only, a in the present case, #@ is an imaginary quantity, since z is always _greater than a*. Making this assumption we find, from the equa- tion of the Sees -5- 1 - 5 = sin’ @, and therefore y = 6 y(— 1) sin @. Hence, if we assume 2 = a cos @, we find y = 6 V(— 1) sin @. (Q. E. D. 331. Or, we may proceed somewhat differently, thus, Let W be the logarithm of = + a which gives ae i at, e Bessie Gm rrtal © ye | * We must regard cos ¢ here as an abbreviation for the expression mee) ON), Indeed we ought to consider the general definitions of | the functions sin ¢ and cos ¢ to be 2 a9 : i : eee cos p = hfe?" Liye $V 1) Aik o=aqcn 1) eo ) ‘from which definitions all the properties of sines and cosines may be very ) readily deduced; as, for instance, the properties | sin? p+ cos’ ¢=1, sin (d+ 9')=sin > cos $'+ cos @ sin #. | It is easy to see from these definitions that, when + is real, cos # and ) sin ¢ are real, and both less than unity: but, when @ is imaginary, cos > is | real, sin ? is imaginary, and cos ¢ is greater than unity. 136 PLANE CO-ORDINATE GEOMETRY. and then, since, by the equation of the hyperbola, (24 oS) (Z- #1, we find aig pot Apmaieh ) mae: oto» o 0 emia lan oi terets an (1) + (2), and (1) - (2) give Ria (eV ON) eat); y=ib (ev —ev), ae To express these formule more concisely, let us put @ V(- 1 for Y, and then, by the exponential values of sin @ and cos }, we have z= a Cos . (5), oy = bE) sino eos: Of course we are as ae at liberty to use the expressions. | (5) and (6) in place of (3) and (4), as we are to employ the | common exponential formule for the sine and cosine of an angle. Properties of the Hyperbola connected with Tangents. Prop. XCIII. 332. To find the angle which the tangent at any point ky of a hyperbola makes with the axis of z. As in Art. 274, we find which determines the angle required.* Hence the equation of the tangent at (hf) is he hy _, ti aah a’ 6? Hence we have C7'= 7? (ay eed 333. Cor. Hence if (az + By = y) be the equation of a tan- gent to the hyperbola, we have the condition 7 y = aa’ — FR’. * The Notes to Arts. 275, 277 apply equally well to the case of the hyperbola, putting — 6? for 6’, PLANE CO-ORDINATE GEOMETRY. 137 Prop. XCIV. 334. To find the normal at any point (AA) of the hyperbola, and the portions it cuts off from the axes. As in Art. 279, we find the equation required to be vk («-h)+¥h(y-k)=0, or Ctr ynasP Hence CGrmer fe h=éh, a aye OF a OC i ee be ote 72 k ae 835. Cor. Hence the normal at P makes equal angles with SP and S'P. For SG = CG -— CS=é¢h —-ae=e.SP, by Art. 321; and S'G = CG + CS=ehiae=e.S'P; therefore SG:S'G=SP:S'P; which shews that the angle SPS’ is bisected by PG, and there- fore that the normal makes equal angles with SP and S'P. 336. Cor. Hence the tangent at P bisects the angle SPS’. This result may be proved geometrically from the property, S'P — SP = 2a, as in the case of the parabola. See Note to \Art. 217. Prop. XCV. $37. To find the locus of the intersection of a tangent of a hyperbola with a perpendicular let fall upon it from either focus. As in Art. 282, we may shew that the locus required is the circle g+y=a* 338. Cor. Hence we may derive a geometrical method of ‘drawing a tangent from, any given point to a hyperbola exactly ‘the same as that explained in Art. 283. i ase * This may be shewn geometrically as follows. (See Note, Art. 283.) Let SY (fig. 101) be the perpendicular from S upon the tangent (¥P); produce SY to meet S’P in Q. Then we have QPY = SPY, and therefore PQ= PS; which gives S'Q=S'P— SP = 2a: and since SC=S'C, SY=QY, we have CY =4S'Q=a. Hence, &c. &c. Pie 138 PLANE CO-ORDINATE GEOMETRY. Prop. XCVI. 339. ‘To determine analytically the tangents of a hyperbola which pass through any given point. & As in Art. 284, we may determine the required tangents fro ay the equation , ohk jae | ae g BRE) € et: Sc 0. 340. Cor. Hence the locus of the point of intersection (AA) of two tangents at right angles to each other is represented by _ W+k=a - 8B’, 4 which is a circle ; unless 6° be greater than a’, in which case the locus is impossible; ¢.e. two tangents cannot be drawn at right angles to each other when 0” is greater than a’. és Prov. XCVII. . 341. If p and p’' be the perpendiculars from S and S" upon any tangent, to shew that pp’ = 0°. q As in Art. 286, we have y eh ae (a a b) SS een ea a ' and therefore pp =—6; 4 the sign of pp’ is negative because the two perpendiculars lie o opposite sides of the tangent. We shall suppose, however, that p and 7’ represent the absolute lengths of the perpendilay without regard to sign, and then we have pp’ = 0’. iliadatals 342. Cor. Asin Art. 287, we have P = = ; and, oltiplyaall | this by pp’ = 8, we find p = O° = BF : Me r+ 2a. 343. Arts. 288-291 hold with so little variation in the case of the hyperbola, that it is unnecessary to repeat them here. ) Properties of the Hyperbola connected with Diameters. Prop. XCVIII. 344, ‘To determine the diameter of a given system oft parallel chords in a hyperbola. M4 PLANE CO-ORDINATE GEOMETRY. 139 As in Art. 292, the equation of the diameter is zx cos 0 sin 0 Tue ws FR Y = 0 . . . ee s s ee * . . CL). a Lb? Art. 293 is equally true for the hyperbola; and the same may be said of Art. 294. As in Art. 295, we may shew that tan @.tan @' = = TERIA Sy Wis Boe was Hence if @ be < 90°, @’ is also < 90°; 2.e. the diameter and its chord make an acute angle with each other. Art. 296 is equally true in the case of the hyperbola, and conjugate diameters are defined exactly as in the case of the - ellipse. 345. Asin Art. 298, if (zy) and (z'y') be any points on two _ conjugate diameters respectively, we have Pror. XCIX. 346. A diameter and its conjugate cannot both meet the hyperbola. If possible let them meet the hyperbola in the points (zy) ‘and (2’y'); then from (3) we have iN 2 aa b° lI oe | | — ae gle SS &.| & { —" ee Ne since (zy) and (z’y') are points on the hyperbola: therefore w+ 2 =a"; which is impossible, since 2° and x” are always each greater than a. Hence only one of two conjugate diameters meets the hyperbola. 347. It is easy to see that if a diameter do not meet the hy- perbola itself, it will meet the conjugate hyperbola. 348. When a diameter does not meet the hyperbola, we shall define the eztremities of that diameter to be the points where it meets the conjugate hyperbola. | + 4 ¢ i aie : 140 PLANE CO-ORDINATE GEOMETRY. 349. Cor. Hence, if (zy) be an extremity (P) of a diameter which meets the hyperbola, and if (2’y') be an extremity (D) of its conjugate diameter, which, as we have shewn, meets the con- jugate hyperbola, but not the hyperbola itself; we have Cine Y zx y i 2 Be Le..eee (1), aie Be | aout (2), and, by Art. 344, = a 250 paul3) Prop. C. 350. To determine the relation between the eccentric angles belonging to the extremities P and D of two conjugate diameters of a hyperbola. If in Art. 348 we assume x =a cos 9, y' = 6 cos @*, (1) and (2) give y=6 V(- 1)sin 9, e =a V(— 1) sin @: which values put in (3), give sin p COS p — Cos ¢ sin ¢' = 0, or sin (@ — ¢') = 0; hence ¢' = @; which is the required relation.t+ 851. Cor. Hence z'=av(- 1)singe= = y, ‘ Op bcos m= Raia S18 L, which formule determine the co-ordinates of D in terms of those of P. Prope OL, 302. ‘The difference of the squares of CP and CD, and the rectangle completed upon CP and CD, are invariable. * We assume y'=6 cos $’, and not «=a cos ¢', because the y' in (2) corresponds to the x in (1), since the axis of y is the possible axis of the conjugate hyperbola. + This may be shewn, without introducing imaginary quantities, as follows. Assume 22r=a(e¥+e%), and .. 2y = b (ev — ev), also Qy' = b (eY 4 eW), and .*. 22’=a (ev - ev); then, substituting these values in (3), Art. 348, we have (ce +4 ev) (ev = ev) — (ef — eV) (ev + eV) =,0, or ew = er which gives WwW’ = w'-y, or W=y, PLANE CO-ORDINATE GEOMETRY. 141 Let CP =r, CD=71', PCX = 0, DCX = 9, (fig. 100); then r=2+y = a cos ¢-'sin’ 9, g”? — a an w= ae a sin” m a b cos” d; therefore Te Tet LO eee ert ete arate a ere (1). Again, if A be the area of the parallelogram completed upon CP and CD, we have A =rvr' sin (6' - 0) =rr' (sin 9’ cos 0 — cos 0’ sin @) = ay — xy = ab (cos’ p + sin’ @). Hence VRE in a> Ope re ti Gi Ck (2). (1) shews that the difference of the squares of CP and CD, and (2) that the area of the parallelogram completed upon CP and CD, are inyariable. 353. Cor. As in the case of the ellipse, we have r=aex — 0b’, y? = &a — a’. $54. Cor. Also SP.S'P = CD’. Prop. CII. 355. 'To find the equation of the hyperbola referred to two conjugate diameters, CP and CD, as co-ordinate axes (fig. 102). Draw any chord QMQ' parallel to DCD; assume CM =z, MQ=y, CP=a, CD= b', Then, by Art. 327, we have PVE Fre Foe en Che Cle But CP =a, MP=2-a, MP’ =x, CD = 6,MQ=y, CP' = — CP, CD’ = - CD, and MQ =- MQ; therefore 2 oh es y° Te ek Oo y or ay ae tts Ty ha 1, a b which is the equation required.” eee * As in the case of the ellipse, by transforming the axes, we may prove that the equation of the hyperbola referred to two conjugate axes is cos? 9 «sin? @\ , cos? = sin’? @’\ , aie 2 x + Tee 2 in 1 ; a’ b a b 142 PLANE CO-ORDINATE GEOMETRY. Hence it appears that the equation of the hyperbola referred | to any two conjugate diameters is of exactly the same form as that referred to CA and CB. ; 356. Arts. 305-308 are true in the case of the hyperbola, if we put — 0’ for J’. Properties of the Hyperbola connected with Asymptotes. Prop. CIII. 357. To determine the asymptotes of a hyperbola. A tangent is said to become an asymptote when the distance © of the point of contact from the origin becomes infinite, provided — at the same time, the tangent itself remains at a finite distance from the origin. j Tf 7 cos 9 and r sin 6 be the co-ordinates of the point of © contact, we have, by the equation of the tangent (Art. 331), and | by the equation of the hyperbola, | 2 COS ame Yast iL AT kh % Gh Gisy Gh eee Gibiy. Cos 02" sin’ Ue 1 yet te cage a Ga es ee (2), (2) shews that, when 7 becomes infinite, cos 6 sin = Tes 3 a b and hence, when 7 becomes infinite, the equation (1) gives us Senedd ra ne =o) SS 0 @eeteeeeeeenesece 3 a” b (8) It appears therefore that, when the point of contact moves to _ an infinite distance from the origin, the equation of the tangent assumes the form (3), which represents two right lines at a — ote hal cos® 8 sin? 6 1 d cos” 8’ ~—s gin? 6’ 1 W ich, since a sgh qr? an hgh iT pean | PLANE CO-ORDINATE GEOMETRY. 143 fi finite distance from the origin: Hence the hyperbola has two asymptotes represented by the equations Tee ees Ss sR SPC gn eee eat Ae Pas? re aa 358. Cor. 1. If we draw HAH" (fig. 103) through A at right angles to 44’, and take AH = AH’ = 3, it is evident that ‘the equations of the lines CH, CH’ are Oe BZ ahah ad —--*=0 = —= 0. Ze eee Hence these lines are the two asymptotes of the hyperbola. Cor. 2. When a= the asymptotes are right angles to each ‘other. In this case the hyperbola is said to be eguzlateral. Prop. CIV. 359. The ellipse and the parabola have no asymptotes. The equation to the ellipse gives Coe Ge sine wert SN at and therefore 7 can never become infinite, since we should then : 72 have tan? @=- =. a Proceeding in the case of the parabola as above, we have z y sin @ = 2m : + COS 0) ee re ae EL), r : 4m sin’ 9 = —— TE UAS S e c (2). r If r become infinite, (2) gives sin @ = 0, and then (1) becomes 0 = 2m, which is absurd ; ¢.e. (1) cannot be satisfied by any ‘finite values of z and y when r = o. Hence, neither the ellipse nor the parabola have asymptotes. Prop. CY. 360. To find the length of a line (7) drawn, at an angle 6 to the axis of z, from any point (A) to either of the asymptotes. _ The equation of the two asymptotes considered as one locus 9° is se yo x 0 {= 144 PLANE CO-ORDINATE GEOMETRY. Hence, proceeding exactly as in Art. 262, we find Ur +2Vr+We=o0O..... (2), & 2 25g when eek 0 eed ae h cos 0 bein a b ye b Wa hig 8°’ which equation determines the required length, giving of course two values of it. 361. Cor. 1. Hence the condition that (Ak) may be the middle point of a line intercepted between the two asymptotes, making an angle 6 with the axis of z, is h cos 6 k sin 0 Z ~ RP = 0 y.. Coon Now this is also the condition (see Art. 325) that (Ak) may ba : the middle point of a chord of the hyperbola making the same angle @ with the axis of z. Hence, if ROPQ'R' (fig. 105) be any line meeting the two asymptotes at R, R’,and the hyperbola at Q, Q', and if P be the middle point of QQ’; P will also be the middle point of RR. q 362. Cor. 2. Hence RQ is always equal to R'Q’, whatever line RQQ'F may be. 363. Cor. 3. (8) gives, tan 0 = 28. hence, by Art. 381, if | (hk) be a point on the hyperbola, the portion of the tangent at (hk) which is intercepted between the asymptotes is bisected at (hk). This may be easily shewn from Cor. 2, by supposing RQPQR to move parallel to itself, till Q and Q’ (and therefore P) coincide. Prop. CVI. 364. To find the equation of the hyperbola referred to its” i asymptotes as co-ordinate axes. ye b 4 Let tan a= fe then the two asymptotes make angles a and —a with the axis of z Hence, by Art. 110, if we put, a oor al Mg i PLANE CO-ORDINATE GEOMETRY. 145 (+) cosa for x,* and (~ x+y) sina for y, in the equation of the hyperbola, viz. x y Ele eS Bh ee A rel sae rise (1), we obtain the required equation. Now, if we assume m?=a°+d, a : b : : we have cos a= 7? sina = aa hence, making the substitu- tions in (1), it becomes (w+ yy} - (ey) =m’, or CY SiS Se Vee es eee ees S02) which is therefore the equation of the hyperbola referred to its _asymptotes as co-ordinate axes. Prop. CVII. 365. ‘To find the equation of the tangent at any point when the asymptotes are the co-ordinate axes. Proceeding as in the note to Art. 275, we have kk and hk= fk! = a“ by (2), last Art., y—-k= ES ey a G ss i) >> a ean). Aye Je Te ahh ‘Hence (3) becomes y-k=- at (x —h); Lh and therefore, reasoning as in the note referred to, the equation of the tangent at (A/) is ri = ke =- = h Mie) 20 Mee os tt) i; : ou: (2 -—h) since m = 4hh, p x y or ein US i Ihe 2h which is the equation required. * This is easily seen from fig. 104, where CH, CH’ are the asymptotes, P any point of the hyperbola, CM’ (=2'), M’'P (=y’) the co-ordinates of P veferred to CH, CH’, and CM (=x), MP (=y) the old co-ordinates. I 146 PLANE CO-ORDINATE GEOMETRY. 366. Cor. 1. Hence the portions, CT, C7" (fig. 106), which the tangent cuts off from the axes, are 2h and 2k; which evi-_ dently shews that 7'7" is bisected at the point of contact P. 367. Cor. 2. Hence the area of the triangle CTT" is inya- riable ; for it is equal to } CT.CT" sin TCT’, which = 2hk sin 2a : Ahkab b a mi: = =— | since sin a = — , CcOSa = — |= ab | since hk = : m m m ye Prop. OVIII. 4 368. The product of the two perpendiculars let fall from any point of the hyperbola upon the asymptotes is invariable. The equations of the asymptotes being on bake ot omg] aD he alee ; the perpendiculars upon them from any point (A) are Lins aR i eta F (is -(3 + 5) and p = “(4 - 7)» Where ot = a So a" i we ri) cw) SH By phe (Ee and therefore Pal Sea | aceon. PP c \a i 2 , if (hk) be a point on the hyperbola. 27.2 ab Hence pp =z «(OE D. oo } g = = Various Problems respecting the Hyperbola. 369. A line AB (fig. 107) is drawn meeting the axes of co-ordinates OX, OY (which are supposed to be oblique) at A and B; if AOB be always a given area, to find the locus of the middle point (P) of AB. | If z, y be the co-ordinates of P, we have OA=2x, OB=2y, and... area AOB = 4zy sin w, where w=ZXOY. Hence, we have 2 tee where c’ is the given area. The locus required is therefore a hyperbola whose asymptot are OX, OY. e Avent PLANE CO-ORDINATE GEOMETRY. 147 6: 370. ‘Lo find the foci of the rectangular hyperbola, xy = mi, from the definition given in Art. 205. _ Let (Ak) be either focus, then we have (cx -hyY+(y-ky =(Axv+ By - CY, and this equation must be identical with the equation zy = m’; we have therefore Meet, Db =-1,-h= AC, k= BC, h’s bP -C’=2ABmn'. From which it is easy to see that h=mv2, k=mv2; orh=-myv2, k=-mvy2. 3871. OX, OY (fig. 107) are two given right lines, and C a given point ; to find the locus of the middle point P of the right line AB, which always passes through C. Let the co-ordinates of C' be A and &, and those of P, x and y; then OA = 2x, OB = 2y, and therefore, by Art. 57, we have h k es “9. Pe 2y or 22y - hy — kx = 0. This equation may be put in the form I ( 4 ( 4 hk ia 3 Z- - Yy —_ — = Re [ 2 2) 2 LY = as If, therefore, we join O and C, and bisect OC in M, the Hicus required is a hyperbola, whose centre is M, and whose asymptotes are parallel to OX and OY. 872. The base of a triangle and the difference of the base angles are given, to find the locus of the vertex. Take the base as axis of z, and its middle point as origin, and let the equations of the sides be , becomes 4 SOA re CU ym (Qt) atekm, acorses (2)): Wet ti be the given difference of the base angles, then we have ASLO a -TaT NG GA Eg aA rere (3); 1+(-m)m 148 PLANE CO-ORDINATE GEOMETRY. 4 and, if we eliminate m and m' from these three equations, the. result will be the equation of the locus required. Hence, subs. stituting the values of m and m’, given by (1) and (2), in (3), we find the equation of the locus to be eae i + tn B (1- ed :)> 2 Z-a x£+a xL- a or 24.2 col Bp vay = ae. Now, when the axes are turned through an angle 6, let this equation become AGS Cy ae ee i lhcb oe ee then (see Art. 190) we have A cos’ 6 + Csin’ 6 = 1, (A — C) cos 0 sin 6 = cot B, A sin’ 0+ C cos’ @=- 1; ; and from these equations we find | A+C=0, (A-G@)cos 20=2, tan 20= cot Pp. : T 1 1 ) Hence 20-5 -P, ap PATICK, > martae 4 and therefore (4) becomes ey =a sin B. Hence the locus required is an equilateral hyperbola whose possible axis makes an angle : - s with the base of the triangle. 373. This problem may be solved more readily as follows. Let the axis of y be inclined at an angle w to the base, the axis of 2 coinciding with it as before, and let 6, ¢ be the angles which the sides make with the axis of z. ‘Then the equations of the sides are pve _ sing | y= as @ 3 0) (x = GA WER (ab Ui sin (w — ) (x 3 a)..{2)8 | and we have (r-0)-o =f. Hence, if we assume the arbitrary angle » to be equal te a — {3, and therefore ¢ = w — 9, (2) becomes | PLANE CO-ORDINATE GEOMETRY. 149 nd (3) x (1) gives ey =a’. dence the locus is an hyperbola, and the axes of co-ordinates wre a pair of conjugate diameters. 374. All the ellipses that may be described on the asymp- otes of a given hyperbola as conjugate axes, and touching the typerbola, have the same area. Let the equation of the hyperbola, the asymptotes being so-ordinate axes, be LY = 1 md the equation of the ellipse 2 2 2 y Tn Sora te Ene Supposing z and y to have the same values in these two 2quations, we have 2 zy ea Parl ye ae a m b md, if the ellipse touch the hyperbola, this latter equation ught to give two equal values of a . Therefore we find UP ee In G? be \. om? fp? or ab’ = 2m’. Now, if A be the area of the ellipse, we have *A = ab, and therefore, since ab = a'b' sin w, A=r7ab' sin w = 2m’ sin o, w being the angle of ordination. Hence A is invariable. 375. To find the locus of the point of intersection of two tangents of a hyperbola which are parallel to two conjugate diameters. If tan‘a be the angle which a tangent, drawn from the point zy) to the hyperbola, makes with the axis of z, we have, by Art. 338, , lay y+ B ie oe Gta 0 va Tat oe Let a and a’ be the two values of a given by this equation, then y° +4 & aa ° fis ay * That 4 = 7ab, will be proved in Art. 391. 150 PLANE CO-ORDINATE GEOMETRY. re But if the two corresponding tangents be parallel to a pai of conjugate diameters, we have i / aa =-- a - pein. A bo Hence y -= =; x -—a a > J or dada fas Foi 2 ieee Se ’ 2a 2b which is the equation of a hyperbola, whose axes are ay2 and 072. Cor. The corresponding locus in the case of an ellipse is 2 2 & Y ae + Se = if Da 26° 376. If two tangents be drawn from any point (Q) of eithe; of the asymptotes of a hyperbola, to shew that the tangent of Li angle they make with each other is | | ab j Pe ee + being the distance of Q from the centre. eS ane CHAPTER XI. MISCELLANEOUS PROPOSITIONS. Prop. CIX. 377 To obtain the relations between the co-ordinates of the extremities of two conjugate diameters in an ellipse or hyperbola | without using the eccentric angle. Let (zy) be P, and (z'y') D; then, as in Art. 298, we have ee yy — -+ — = 0 Sievieh Gann 6 or 610 0) 0) 8. Cie Ne 1 e Qa 6° (1) “an y Let us assume that — = u ie being some unknown quan- a | tity, then (1) gives a — U = . Now, by the equation of the ellipse, we have (2) (4 ) SN eI 3 atk a _ pl ee Nae nee Jw (E+ B) a1, and @. #@=1, or vw=+ 1. Hence we have x’ x _Y Ee 0g Ebel 0. aun a b which are the relations required. The double signs refer to the two extremities of each diame- ter: it is usual to suppose P and D to be the extremities to which the upper signs belong: on this supposition we have a i ee (a as 206. Ole ie ss . +» (3) 162 PLANE CO-ORDINATE GEOMETRY. ¢ In the case of the hyperbola we have Eee eS a ey 5 ; ze ee : = Assume ~ = U a and .. phe then, since (2’y’) is a point of the conjugate hyperbola, we have which, since (xy) is a point on the hyperbola itself, gives «7 = 1, | and ..«=+ 1, as before. Hence we have | or, omitting the upper signs, ey Come es : 0 Sarre), ae Oceed oi Prop. Gix. 378. From the preceding results to deduce the properties of | conjugate diameters. | From (2) and (3), we have | 2 ¥ : pe ety =a 2 +B b a 2 2 2 eo pe ata afi) ee oe oe ce +y =a + (2 +y’), or CD’ + CP? = a* + 8. Again, from (2) and (3), we have ay —xy = ab. PLANE CO-ORDINATE GEOMETRY. 153 In this result, putting r cos 0, r sin @ for z, y; and’ cos @, r sin 0’ for z', y'; we have rr sin (0 - @) = ad, t.e. the area of the parallelogram completed upon CP and CD ‘is invariable. In the case of the hyperbola we may shew, in exactly the same way, that a2? +" = a? J.B - or CP*- CD = a - & , Bae 2 OG Faye Again 1A Bae saat faa a iy a : i) xy —-xy=ab; and hence rr’ sin (0 — 0) = ab. Prop, CXI. 379. A conic section being referred to any axes of co-ordi- nates originating at the centre ; to determine the radius of the circle, having its centre at the origin, which touches the conic section. The general equation of a conic section, referred to the centre as origin, is Arete Diy Cu Manes eats areal yg Let the equation of the circle, whose radius we wish to deter- mine, be oie fee Oi Ara anbioon Ae ea (1) - : (2) gives (4- 1) # + 2Bry +( C- Y= 0-8). In general this equation gives two different values of - ; but these values will become equal when the circle touches the 154 PLANE CO-ORDINATE GEOMETRY. conic section, therefore the first member of the equation will be — a perfect square, and we shall have . ial febsee re or 5 (4+0)4 + AC — 3 = es ee (4), which equation, therefore, eee the radius of the circle — required. 380. Cor. 1. When a circle, concentric with a conic section, _ touches it, the points of contact must evidently be at the ex- — tremities of either the major or minor axis (or the extremities _ of the possible axis in the case of a hyperbola): hence the © values of 7 given by (4) are the semiaxes, @ and 8, of the conic — section ; we have, therefore, 5+ G7 446, ap AC- B Cor. 2. Hence, if AC be > B’ and A + C>0, a’ and 8 are a both positive, and therefore the curve is an ellipse ; but if AC be < B either a or 6° is negative, and the curve is therefore a _ hyperbola. ' b Prop. CXII. | 381. To determine the position of the axes of the conic — section in the preceding proposition. g Supposing the first member of (8) to be a perfect square, we have . {4-abe+ By=o, {e-shy+ Be=0; and, multiplying the first of these equations by y, and the 3 second by z, and subtracting, we find ‘ (A-C)ay+ BY’ -2#)=0: and, putting z = 7 cos 0, y =r sin 0, we obtain 2B an 20 ee an 26 Rae * For if La* + 2Maxy + Ny’ be a perfect psnae its square root is either VL. Be sch y, or J/N.y + TL x L phate oat PLANE CO-ORDINATE GEOMETRY. 155 which determines two values of 0, whose difference is 90°, and these values are the angles which the two axes of the conic section make with the axis of wz. Prop. CXIII. 382. 'To determine the magnitude and position of the axes of the conic section represented by Az’ + 2Bry + Cy’ = 1, when the co-ordinates are oblique, the angle of ordination being w. We may proceed exactly as in the preceding proposition, only instead of the equation (2) we now have z+ Izy cosa+y =17; and therefore, instead of (3), we have w A eer : (4-7) e+ 2(B-% Jay +(0- ss) 9 = O-n(5) ced ulate Wawel > 7 r 1 A+C-2Bcosw 1 AC-B or Se a i _- Seen cera Ee ew 0. r sin® w - mn Hence - + i. A+C-2Bcosw I > AC it ‘oa sin’ w me 0. Sinan Again, to determine the position of the axes, we obtain from (5), as in Prop. cXI1., Ax + By -(«+y cos) = =0, Cy + Bx - (y + & COs w) and... (Ax + By) (y+ cos w) -(Cy + Br) (x+y cos w)= 0, or (A cos w — B) 2 +(A-C) zy +(B- Coosw)y’ = 0. This equation gives two values of e re ns suppose, and we have, therefore, either 7g pe-gqy=0, or pxr-gy=0, which are evidently the equations of the two axes. 383. Cor. If we suppose A = “4 j ae iF? B=0, the equation of the conic section becomes ci 2 gtd ae b” 156 PLANE CO-ORDINATE GEOMETRY. and (5) becomes 4 2 és 1) 2-2 eo w.ay+(7-1)y=0, Y" 2 and .". (3-2) (Fa- 1) = costo, or rm — (a? + 6”) 7 + ab” sin’ w = 0. Hence a’ +6" =a'+6? and ab'sin w= ab, which are the two properties of conjugate diameters previously obtained by different methods. Prop. CXIV. 384. A tangent is drawn from a given point (zy) to an ellipse, to find the angle (Y) which the portion of it between (zy) and the point of contact (A%) subtends at either focus. Let r and r’ be the distances of (zy) and (hk) from the focus S’; then we evidently have _“-ae h-ae yk cos p= 2 pie Hb lea are (1), and se + 2 ow) SEEN eee. (2). Substituting in (1) the value of ky obtained from (2), we have 2 rr cos b = (x - ae) (h - ae) + B - 2 hz = 2(eéh-ae)-aeh+a@, since =a (1- oh (a — ex)(a—eh): and hence, since 7’ = a- eh, we have a— ex cos w= ‘ r which determines the required angle. 385. Cor. 1. If Q (fig. 108) be the point (zy), P the point (hk), and QRM an ordinate ; then S'Q=r and S'R=a- ex; hence cos PS'Q = =o : and hence, if we draw QT perpendicular to S’P, we have DS Leon fe. PLANE CO-ORDINATE GEOMETRY. 157 886. Cor. 2. If QP’ be the other tangent drawn from Q, we pcan G7 OG , have cos QS'P’ = . Hence, if from any point Q two tan- 7 gents, QP, QP’, be drawn to an ellipse, the lmes QP and QP’ subtend equal angles at either focus. 387. Cor. 8. The angle y being given to find the locus of Q. We have r cos p=a- ex, or, transferring the origin to S", rcosW=a-—e(«+ae)=a(1 -e)- ex; 2 which, introducing polar co-ordinates, and putting ea =l, os e = =e becomes z cos = ——__——__ ; 1 +e cos 0’ which is the equation of the locus required; it is therefore a conic section having its focus at S’, and its eccentricity and latus l rectum equal to °_ and cos W cos ib Prop. CXYV. 888. Having given the magnitude of position of two conjugate diameters of an ellipse, to find the magnitude and position of the axes. respectively. Let 7, 7’ be the two given conjugate semi-diameters, w the angle they make with each other, and 6, 6’ the angles they make respectively with the axis major. Then we have PA Te er le Og criss oe coos iets GE ToUN GIN) FLL Peas eae it Son eee Uo therefore, adding and subtracting, and putting, for brevity, ° . Ps 2 r+2rr sinw +r? =U, yf —2rr' sinw+7r =0’, we have * at+b=u4tzY, and .. 2@=%U+2, 9%=u-v; which determine a and 0 in terms of 7, 7” and w. To find 9 we have, since 0' = 6 + w, 2 tan @ tan (0 +4) =~ “a, which determines 06. 158 PLANE CO-ORDINATE GEOMETRY. Prop. CXVI. 389. To find the magnitude and position of the two equal conjugate diameters. Using the notation in the preceding proposition, and putting r = 7, we have, by (1) and (2), x) et & = 5 - sin 2ab QO = 32 ian ¢ which determine 7 and w. ‘To determine @ we have 2 2 eee) ry? cos’ § 7* sin’ 0 : ERM or koe a 1 = cos’ 8 + sin’ 6; a b 97" - 2a? D710" ——,— cos’ 9 + —_,— sin’ 6 = 0, a ve or 08". 0+ 7 sin’ 8 = 0, since 277 =a’ + 0’. a b ; 4 Pye. Hence tan 6 =+-, which determines the position of the two a . diameters. Prov. CXVII. 390. To find the greatest and least values of w. : ab ‘ ; : We have sin » =—,, and therefore sin w is a maximum or rr minimum according as 77’, or (77r')’ is a maximum or minimum. sip a’ + b’\’ Cre Now rr? =r (+ -97) = — (a? . 9 D) 3 ae + BD? which is evidently greatest when 7? =“ : , and therefore Pies te he aa ; | > = z also, and least when 7” has its greatest value a’. Hence sin w is least when “r = r’, and greatest when r= a; and . T ° therefore, since we suppose w to be greater than 5?” is greatest when 7 = 7”, and least when 7 = a. eee Se IL . Hence the minimum value of w is a0 and the maximum yalue is given by the equation 2ab sIn w = : “ae et is PLANE CO-ORDINATE GEOMETRY. 159 Prop. CXVIII. 391. To find the area of an ellipse. If we inscribe in the elliptic quadrant ABC, (fig. 109) the Pet IE rectilineal figure pqp'q'p'q’......3 PG py, &c. being parallel to CA, and gp’, q'p, &c. | pape sere to CA; it is clear that we may make the area of the rectilineal figure differ from that of the elliptic quadrant as little as we please, by taking each of the sides pq, pq’, &c. sufficiently small. Let (xy), (vy’), (2'y’), &c. be the points q, 9’, g’, &c., X the area of the rectilineal figure, and A that of the ellipse: then we have X=y(e#-0)+y @-ax)t+y(e-2v)+ &e.; and if X' be the corresponding rectilineal figure inscribed in the circular quadrant ACB’, we have, by Art. 261, = ; y («@-0)+ ; y (e = 2) +5 y (a2) + &e: septs ah XxX’. a Hence oy: may be made to differ from A as little as we a please: but X' may be made to differ from the area of ABC, 2.€. a , as little as we please; therefore the difference between b ra ; A and - = must be less than any small quantity we choose to a specify: which cannot be, unless 4 = — ba Therefore A = x and hence the area of the whole ellipse is za. 392. Cor. 1. Hence, if we produce the ordinate MP (fig. 110) to meet the circle described on -4.A’ as diameter in Q, it is clear that aes, ot UP = : area. A’ MQ. Also area CMP = shal area CMQ = — area CMQ, MQ a area CPA’ = : area CQA’ Ria , if 2 QOA'=4; 160 PLANE CO-ORDINATE’ GEOMETRY. hence the expression for the area CPA’ is 5 pab. 393. Cor. 2. If we put bv(-1) for 4, and y for pv(-1), we have area CPA’ = 43 ab; which is the expression for the hyperbolic area CPA’ (fig. 111), * ners W = log (< + a » (see Art. 830). ab Prov. CXIX. 394. To find the area of the portion AMP of a parabola (fig. 112), MP being any ordinate. Draw PN parallel to AM, let p, p' be any two points of the arc AP whose co-ordinates are zy, zy’, and draw q'pm, p'qm' parallel to AN, and gpn, p'q'n’ parallel to AM. Then, if we © denote the areas mpp'm' and npp'n' by X and Y, we have X = y(2' - z)+ area pp'g, Y=x(y'— y)+ area py”. y(z — 2)% Hence it is clear that 2 may be made to differ fron. xy - y) as little as we please by bringing p’ up to p. Now, since y= 4mz, — and y = 4mz', we have a(y'-y) 4mx y'-y y + 4 ; and he may be made to differ from 2 as little as we please _ by bringing p’ up to p. It follows therefore that = may be | = Rae made to differ from 2 as little as we please, by bringing p’ up top. — Now we may divide the area AMP into a series of portions — such as X, and the area ANP into another series of portions — such as Y, and by what has been proved, we may make the ratio of each of the former series to the corresponding one of the | latter differ from 2 as little as we please. Hence the ratio of the — area AMP to the area ANP must differ from 2 by a quantity — less than any small quantity we choose to specify: which cannot — be unless area AMP = 2 area ANP, or, what is the same thing, — area AMP = 2 area AMPN. Hence it appears, that the area of the parabolic segment 4 PM — is two thirds of the circumscribing rectangle AMPN. Bee res] i PLANE CO-ORDINATE GEOMETRY. 161 Prov. CXX. 395. Two chords, drawn from the extremities of the major | axis to any point of an ellipse, are always parallel to a pair of conjugate diameters. Let (4k) be the point to which the chords are drawn, then the equations of the chords are “-a_y “+a iy Ad uhhh hee th Hence, if 0 and 6’ be the angles which these two lines make with the axis of x, we have V5; \ k: tanf=——_, tan@’-= 4 2a A+a@ Vind and). ictani@ tan: Oo = —-— li ale 6’ y(gfaty ve = — a» since | + iy a iis which shews that the two chords are parallel to a pair of conju- ‘gate diameters. 396. Cor. In the same way we may shew that, if two chords be drawn from the extremities of any diameter to any point of an ellipse, they are parallel to a pair of conjugate dia- ; sin 6 meters. Only instead of tan 6 and tan 9’ we must write Oe) , Oi an BO and @ and b' for a and 6. We also assume that 1 if 9 and 6’ be the angles which any pair of conjugate axes make with the axis of z, when the ellipse is referred to a pair of con- jugate axes, then sin 6 BLOG e. ih UO. sin (w — 8) sin (w — 6’) aes This may be proved, as in the case of the ellipse referred to : sin (w — 0 the axis major and minor, by putting h+r femiona ed | ; a ee or i for x and y in the equation — + J =1; and then sin @ eid e putting the coefficient of 7, in the result, equal to zero. M RS CHAPTER XII. OF THE GENERAL EQUATION OF THE SECOND DEGREE BETWEEN & AND ¥, 397. In the following propositions we shall consider the equation of the second degree in its most general form Az? + 2Bry + Cy? + 2Dx+ 2Hy + F= 0. We have already (in Chap. vit.) investigated the nature of the locus represented by this equation, and proved that it represents — in general an ellipse, parabola, hyperbola, two right lines, one right line, or an isolated point. We have also given a method of determining the nature and position of the locus when the coefficients A, B, C, &c. are given; and shewed that this method is simplified by first reducing the equation to the form Ax’ + 2Bry + Oy’ + F = 0, i.e. by transferring the origin to the centre, which may always be done, except when B’- AC =0. We shall now pursue this subject more into detail. Prov. CXXI. 398. To determine the length (7) of a right line drawn, at an angle @ to the axis of x, from a point (A/) to the locus repre- sented by the equation Ag? + 2Bary + Cy’ + 2Dx + 2Hy + F=0. As in Arts. 68, 128, &c. put h +r cos 6 and &+,r sin @ for z and y in the equation of the locus ‘ Ag? + 2Bry + Cy’ +2Dx+ 2Fy+ F=0....(); and it becomes Ors 2VrtW = 00... ope PLANE CO-ORDINATE GEOMETRY. 163 where U = A cos’ 6+ 2B cos 6 sin 0 + C'sin® 0 = (Ah + Bk + D) cos 0+ (Ck + Bh + E) sin , (33 W = Ah’ + 2Bhk + Cl? + 2Dh+2EK + F (2) determines the required distance 7, and in general gives two values of it. Using the notation in Arts. 129, 211, &c. we have PQ+PQ =- Z| PQ*POQOe= we 399. Cor. From the equation (4) we may, just as in Arts. oy 132, 133, 185, &c., draw the following conclusions : (1) That the fing (hk) bisects the chord drawn through it at an angle @ to the axis of 2, when V = 0, 7.e. when | (Ah + BE + D)cos 0+ (Bk + Ch + E) sin 0 = 0... (5). _ (2) That the equation of the diameter of the system of _ chords inclined at an angle @ to the axis of z, is ; (Az + By + D) cos 6+ (Cy + Bu + E) sin 0 = 0, or (Acos6+Bsin 6)z+(Csin 0+Bcos0)y+Dcos 0+ Esin6=0...(6). (8) That the angle (0) which the tangent, at the point (24) of the locus, makes with the axis of z, is given by the equation = 0; or Ah + Bk + D tan § = — CEE R ILO one wwe ia ss a wire Cay _ (4) That hence the equation of the tangent at the point (AA) as” (4h + Bk + D) (2h) +(Ch+ Bh+ E)(y-&)= 0, ——— $$$ ia: * We may find the equation of the tangent in the following manner also, as in the Notes to Articles 214, 275. Let (hk) and (Wk) be two points of the locus; then Ah’ + 2Bhk + Ch? + 2Dh+ 2H + F =O oo....000e (1). Ah® + 2Bhk' + Ck? + 2Dh' + 2EK + F'=0 .0.. cece (2) _ Subtracting (1) from (2), and observing that Wh'—hk=Wk' -Wk+ Wh -hk=h' (k'-k)+k(h'—hy), we have, dividing by h'—h, A (h'+h)+2Bke+2D+{C(k' +k) + 2Bhi+ 2H} 7 nok =) eae 164 PLANE CO-ORDINATE GEOMETRY. which, in virtue of the equation (1), since (A/) is a point of the — locus, becomes ; (Ah + Bh + D)2+(Ck+ Bh+ E)y+Dh+ Ek+ F=0...(8). — (5) That hence the equation of the line joing the points of contact of the two tangents drawn from any point (4/) to the © locus, is » 4 (Ac + By + Dyh+ (Oy + Be+ E)h+ De+By+F=0, | or (Ah+ Bk+ D)x+(Ck+ Bh+ E)y+ Dh+ Ek+ F=0... (9). : (6) That this last equation is also the equation of the locus of the point of intersection of the pair of tangents drawn at the — extremities of any chord passing through the point (44). (7) From the second of the equations (4) we may shew, just — as in Arts. 213, &c., that, if QQ’, RR’ be two chords of the locus, ~ and P their point of intersection, the ratio PQ.PQ': PR.PR . is not altered by moving the chords parallel to themselves in — any manner. Prop. CXXII. 400 To find the centre of the locus represented by the general equation of the second degree. The centre is that point which bisects all the chords drawn — through it; therefore, if (2k) be the centre, the equation (5), Art. 399, must be true for all values of 6. Hence we have 4 Ah+ Bki+ D=0....0), Ch + Bh+ Bb =0. Oe which two equations determine / and /, and therefore the centre of the locus. If we actually calculate 4 and & from these equations, ee a eh Cocca Om Vig eee HC ed Oe ae Cor. 1. If AC - B’=0, h and k are in general infinite, 2.e. the locus has no centre. We may also see this from the a a Hence the equation of the line joining (A/) and (/’F’) is A (h'+h)+2Bk+2D GEN: C(k'+k)+2Bh' +2E " which, when A’ and #' approach / and k, becomes in the limit Ah+ Bk+D ¥-*-— cee mi Re” which is the equation of the tangent obtained in the text. y-k=- PLANE CO-ORDINATE GEOMETRY. 165 equations (1) and (2), Art. 400 ; for, when B’ = AC, they become . Jie tt C= — (pe ing 7) Ah+ Bik + D=0, Ah + Bh+ = =o, , EA which equations are manifestly inconsistent, unless D = BR? and therefore cannot be satisfied by finite values of & and 4. Consequently there can be no centre when B* = AC, except in EA a: 401. Cor. 2. In the particular case just mentioned, the equations (1) and (2) become identical, and therefore equivalent to only one equation between 4 and &. Consequently an infinite number of values of and & will satisfy the conditions (1) and (2),* and therefore the locus will have an infinite num- ber of centres, whose co-ordinates are restricted to satisfy the equation Ai + Bk + D=0; 7.e. all these centres are situated on the right line represented by Az + By + D=0. We may verify this result in the following manner. If B= AC, BD = EA, and .. BE = CD, the general equation of the second degree becomes ae e+ 2Bary + Ao yf + 2Dz + 2By + F=0, the particular case where D = or — (Dz + Eyy + 2(Dx + Ey) + F= 0. If M and WN be the two values of Dx + Ey given by this equation, we have (Dz +.Ey —- M)(Dz + Ey - N)= 9. Hence the general equation represents two parallel right lines. Now it is evident that any point of the right line, drawn parallel to these two lines, and equidistant from them, is a centre of the two lines considered as one locus. Hence there are an infinite number of centres all situated im a right line. * If B= AC, and BD=AE, then BE= CD also, and the expressions rh 0 : for h and &, instead of being infinite, assume the form 5) which shews that h and & are indeterminate. 166 PLANE CO-ORDINATE GEOMETRY. Prop. CXXITI. S 3 402. To find what chords of the locus, represented by the — general equation of the second degree, are perpendicular to _ their diameters. By equation (6), Art. 399, if 6 be the angle which a chord makes with the axis of z, the angle (6@') which its diameter makes with the axis of z, is given By the. equation Acos 0+ Bsin 0. Csin 0+ B cos 0’ and if the diameter and chord be at right angles, we have tan @' = - cot 6. Hence (C'sin 6+ B cos #) cos 6 =(A cos 6+ B sin 8) sin 6, tan @' = ands a B cos’ @ — sin’ #) = (A — C) sin 8 cos 8, or tan 20 = ar which equation gives two values of @ differing by 90° from each — other. It is the same equation as that obtained in Art. 187 for — determining the position of the axes of the locus. Hence there — are only two diameters, namely the axes, whose chords are at ‘ right angles to them. Prop. CXXIV. 403. 'To shew that the locus represented by the equation of © the second degree is in general an ellipse, parabola, or hyper- — bola, according as B’ is less than, equal to, or greater than, AC. © Recurring to the equations (3), (4), Art. 398, we find that, in order to make P the middle point of every chord which passes i through P, we must have PQ =-— PQ’, or V = 0, for all values — of #, and therefore i Ah+ Bk+ D=0, Ck+ Bh+ E= 0. Now these equations are inconsistent when % = A , t.e. When — Be AC [except =: Z also | . Hence, in general, when | B’ = AC, there is no point which bisects all the chords drawn through it ; and consequently the locus has no centre; @.e. it must be a parabola. PLANE CO-ORDINATE GEOMETRY. 167 But if B® be not equal to AC, we may always determine h and & from the two equations just put down; therefore the locus has a centre, and it must be, in general, an ellipse or hyperbola. To distinguish which it is, we have only to suppose that QQ’ =o, which can only be the case in a hyperbola, and therefore that PQ. PQ' =a; which gives U = 0, or A cos’ 0+ 2B cos @ sin 0+ Csin’ 0 = 0, Bap AC) pent ariea eo Now this value of @ is possible only when B’ is > AC; and therefore the locus in an ellipse or hyperbola, according as B is mor > 4 C.* and .. tan @ = — * We may obtain this result somewhat differently, as follows: For each value of x two values of y are given by the quadratic equation Cy? +2(Ba+ FE) y+ Av’ +2Dze+ F=0; and these values are real, provided (Bz + E)*? be > C(Ax?+2Dzr+F), or (B?- AC) +2 (BE-CD) ~ + (E*- CF) Ze 0. Now, by assuming z to be a sufficiently large positive or negative quan- tity, we may make the last two terms of the first side of this inequality as small as we please, and therefore the first side to have the same sign as B?-AC Hence, if B® be < AG, the inequality is not satisfied by very large positive or negative values of x; but, if B? be > AC, the equation is satisfied by any very large positive or negative values of x. If B’= AC, the condition that y may be possible becomes V2 x 12 (He CODY + > 05 and therefore, if BE— CD be positive, y is possible for very large values ob e: provided they be positive; and if BE- CD be negative, y is possible for very large values of , provided they be negative. Exactly the same reasoning will apply if we solve the general equation for x instead of y. Hence the locus has no points indefinitely distant from the axes of co- ordinates when B? is less than 4; it has points indefinitely distant from the axes on both sides of each of them when B? is greater than 4C; and it has points indefinitely distant from the axes, but only on one side of each of them when B? is equal to AC. Therefore the form of the locus corresponds with that of an ellipse, parabola, or hyperbola, according as B? is less than, equal to, or greater than 4C. 168 PLANE CO-ORDINATE GEOMETRY. Prop. CXXV. | 404. To determine under what circumstances the locus ; represented by the general equation of the second degree is two _ parallel right lines. : If the locus be two parallel right lines, every point of the © right line drawn parallel to, and equidistant from them, is a centre, and therefore the locus has an infinite number of centres. Consequently the equations Ah+ Bk+D= 0, Ch+ Bhi E=0, must be satisfied by an infinite number of different values of h and &; which cannot be unless the two equations be identical. — We have, therefore, sin Aired cn Bink sate se or AC = B’, and BE= CD (or EA = BD). If these conditions hold, the general equation becomes aS w+ 2Bary + By 2Dzx2+ 2Hy + F=0, B , 7 or Rp (B+ Fy) + 2 (De + Hy) + F=0, B ba / BF or gp P2+ By +12 [(1- Fs) =o. If BF be < ED, this equation represents two parallel right lines ; but, if BF’ be > ED, it represents no locus. Hence the conditions necessary in order that the general equation of the second degree may represent two parallel right lines, are 4 ep - — = and BF < ED. Prop. CX XVI. 405. To determine the circumstances under which the general equation of the second degree represents two intersecting right lines, or an isolated point. When the locus is two intersecting right lines, or an isolated point, the centre is manifestly a point of the locus. Now, if we transfer the origin to the centre, the general equation becomes Ax’ + 2Bry + Cy’ + Dh+ Ek + F=0......(1), PLANE CO-ORDINATE GEOMETRY. 169 (see Art. 193 dss) where / and & are given by the equations Ah+ Bk+ D=0....(2), Ck+ Bh+ E=0....(8). Hence, since the centre, which is now the origin, is a point of the locus, (1) must be satisfied by the values x=0, y=0; ‘therefore we have Dh+ Eki F=0..............(4) and AG 2 Bay + Cy ='00..': B+v(B- AC) or OAS ay ee Yie200 2 73.109 Say Also, if we substitute the values of h and 4, got from (2) and (8) in (4), we find CD’ - 2BDE+ AE’+ F(B’- AC)=0....(6); (6) is therefore the condition necessary in order that the locus may be two intersecting right lines or an isolated point. If B’ be > AC, (5) represents two right lines intersecting at the ‘centre; but if B’ be < AC, (5) is satisfied only by z = 0 and y= 0, and therefore represents an isolated point, namely the centre. Hence the locus will be two intersecting right lines, or an isolated point, according as B’ is greater or less than AC. Prop. CX XVII. 406. To determine whether the locus represented by the general equation of the second degree has asymptotes ; and if so, to find their equations. _ If we put 7 cos 6 and r sin @ for hand &, in the general equation, 47? 4 2Bhk + Ck? +2Dh+2Ek+ F=0, and, in the equation of the tangent, (Ah+ Bk + D)x+(Ch+ Bh+ E)y+Dh+ Ek + F=0, they become A cos 0+ 2B cos 6 sin 6+ C'sin® 0+ : (D cos 6+ Esin 0) + = 0, and (Ax + By + D) cos 0+ (Cy + Br + B)sin 0+~ =0, Hence, if 7 become infinite, z.e. if the point of contact be at an ‘infinite distance from the origin, we have | A cos? 0+ 2B cos @ sin 0+ Csin’ @= 0, and (Ax + By + D) cos 0+(Cy + Bu + E)sin 0= 0. . 170 PLANE CO-ORDINATE GEOMETRY. The first of these equations gives A cos 0+{Biv(B’- AC)} sin 0=0, ¢ and, thence, the second becomes 4 (Ax + By + D) {BBtv B’- AC)} -(Cy+ Br+ £)A=0...(1), : which is, in general, the equation of the tangent when the point f of contact is at an infinite distance from the origin. ‘ Hence, when B’ is > AC, the locus has two asymptotes, 4 3 3 tb represented by (1); and when B’ is < AC, it has no asymptote. — If B’ = AC, (1) becomes F 0.2+0.y+DB-AEH=0, which shews that the tangent goes off to an infinite distance from — the origin, for this equation cannot be satisfied by any finite ; values of 2 and y.* ‘Therefore the locus has, in general, no 4 asymptotes when B’= AC. 407. Cor. It is evident, from the form of the equation (1), that the two asymptotes intersect at the centre ; for the co-ordi- . nates of the centre satisfy the equations Az+ By+ D=0, Cy + De+ b= 0. Prov. CX XVIII. ; 408. To obtain the equations of the asymptotes indepen-— dently of the equation of the tangent. i Assuming the notation in Art. 398, we have Ur? +2Vr+w=0, A 1 or U+2V. —-4+W. = =09. r r Now if U=0 and V = 0, this equation makes each of the two- 1 ; values of equal to zero: hence, if U=0 and V=0, the two points, where the line, drawn from the point (2/) at an angle 0 to the axis of z, meets the locus, are at an infinite distance from (hk); under which circumstances this line is evidently an asymp-_ tote. Hence the equations U=0 and V=0 make (hk) a point on an asymptote of the locus. If, therefore, we eliminate 0 from” * Except when DB-—AF=0, in which case the locus is two parallel right lines. PLANE CO-ORDINATE GEOMETRY. 171 these two equations, we find a general relation between the co-ordinates, h and 4, of any point on an asymptote of the locus. Hence, putting for U and V their values, and substi- tuting z and y for h and k, the asymptotes are determined by eliminating @ from the equations A cos’ 0+ 2B sin @ cos 6+ C sin’ 6 = 0, (Az + By + D) cos 0+ (Cy+ Br + EF) sin 0 = 0, which agrees with the preceding proposition. Prop. CX XIX. 409. 'l'o shew that ifthe equation of the second degree be put in the form (az + By + y) (ax + B'y + y') =n, the asymptotes are represented by the equations, az + By+y =0,anda'x+ B'y+y'=0. Putting h+~r cos 0, and k+~7 sin 6, for z and y, the equation of the locus becomes 72 4 ofp 4 We 0, where U=(acos @ +3 sin 6) (a’ cos 6+ 3’ sin 8), V=(acos 0+ (sin 6) (a’h+'k+y')+(a'cos 0+ /3'sin 8)(ak+ Bh+y)=0. Lo obtain the equations of the asymptotes, we have only to put U=0, and V = 0, and eliminate 0 from these two equations. Now U= 0 gives acos 0+ (§ sin 0=0, ora’ cos 0+ (3' sin 0=0; and therefore V = 0 gives ak + Bk +y=0, ordh+Bhk+y =0. Hence, putting x and y for 4 and &, the equations of the two asymptotes are ax + By + y= 0, ax+By+y =0. Prop. CXXX. 410. To put the general equation of the second degree in the form (az + By + y)(a'z + By + y') =n. Transferring the origin to the centre, the general equation of the second degree becomes (see Art. 193, dzs) Az’ + 2Bry + Cy’ + Dh + Hk +F=0.... (1); h and &, the co-ordinates of the centre, being given by the equa- tions Ah Be} D=0, (Ch Bank = 0. : gd ange igh Cs Now Az’ + 2Bry + Cy’ = Ay iatigetat ara) 1772 PLANE CO-ORDINATE GEOMETRY. when Xd and pare the roots of 2° + gn 2 + 5 = 0. Hence (1) becomes A (2 — dy) (a - wy) + Dh+ Ek+ F=0; Now transferring the origin back to its old position, 2. e. putting z-hand y - k for x and y, this equation becomes Afx-hry+h-rk\ {a-pyth-pk}+ Dh+ Ek+ F=0, which is the equation (1) put in the required form, h and k_ being given by the equations Ah+ Bk+ D=0, Ckh+ Bh+ H=0, and d, « being the roots of Az’ + 2Bz+ C= 0. Cor. Hence the equations of the asymptotes are Gonyeng pace NGon ny See Prop. CX XXII. | 411. To trace the form of the locus represented by the gene- — ral equation of the second degree, by solving the equation for y. — The general equation of the second degree may be put in the form A Be+k Az’ + 2Dz2+F Yn a gig 4 Aine Caan mmeaie which gives _ Bz +E V{(B’- AC)2?+2(BE- DC)x+ E’-CF Cs C ; Ba+ re + R suppose, Let AB (fig. 113) be the right line represented by the equa- tion Cy + Bx + E= 0; then, if we take OM = z, and draw the ordinate MWPQ to meet AB in P, it is evident that . Bark MP =- C Hence, if we take PQ = R, and PQ’ = - R, we have Men uo -_P*tf_R It follows therefore that Q and Q' are the points where the ordinate drawn from M meets the locus, and this is generally true for all positions of M@. Also, since PQ = PQ’, it appears — where Wh = a ee R is impossible for all values of z less than — = PLANE CO-ORDINATE GEOMETRY. 173 that the line 4B is the diameter of the chords of the locus which are parallel to the axis of y. To make out how the locus lies with respect to 4B, we have _ only to trace the values of R corresponding to different values of x in the following manner. Ist. Let B’ = AC; then we have C?R? = 2Mz +N; assuming, for brevity, M= BE- DC, N= E’ - CF. If M be positive, 2Mx+ N is positive for all values of z negative for all others, and continually in- N eater than — — greater than We N creases from 0 to © , when z increases from — oi to co. Hence , possible for all greater values, and increases continually from 0 too, when z increases from — — to «. It follows therefore that (fig. 114) represents the locus, where OM, = - ay Except when BE - DC, or M, is zero, in which case & is invariable, being real when N, or £’- CF, is greater than zero, and impossible when less. Therefore the locus 1s two right lines parallel to AB, or has no existence, according as EY’ is,> or < GF. ond. Let B? - AC be negative ; then we have CR =(B- AC){z’ +2Mzx+ N}, ea! Meer OT B- AC’ iB AC’ therefore R’ has the same sign as, and is proportional to, _(¢ +2M2+N) or M*-N-(@- MY’. Hence, if M? be less than N, R’ is essentially negative, and therefore there is no locus: but if M? be greater than N, R’ 1s positive for all values of x which make (a — M) less than M*—N, ; c.for all values of x between M - v(M?-N), and M+v(M"-N), and negative for all other values: also, R’ is zero when a= M+iv(M’? -— N), and is greatest when z= MM. Hence it 174 PLANE CO-ORDINATE GEOMETRY. appears that fig. 115 represents the locus, where OM, and OM, are M+ vV(M- N), and OM, = M. ; If M* = N, the only real value of # is obtained by putting z' = M, and therefore the locus is a single point. | 3rd. Let B’ — AC be positive, then Ff’ has the same sign as, and is proportional to | 2+2Mx+N or («- M+ N- M’. Hence, if M” be less than NV, R’ is essentially positive for all values of z, and is least when x= MM. ‘Therefore (fig. 116) represents the locus. But if M’ be greater than N, R is negative for all values of x between M-\(M’- N) and M+(M’— N), and positive for all other values; also, #’ is zero when z = M+ /(M’* — N), and continually increases, when © (x — M) increases from M’- Nto». ‘Therefore (fig. 117) re- presents the locus. | If M’ = N, CR’ =(« —- MY, and therefore . Baw a M Suh Crt eee which represents two right lines. Thus we have traced the form of the locus in all the cases that may occur, and the results thus obtained agree exactly with those previously obtained by different methods. Of the modifications to be made in the preceding propositions when the co-ordinates are oblique. 412. In Prop. cxxi., instead of putting z=h+7 cos 0, y=k+rsin 0, we must putz=h+rm, y =k + rn, where sin (w — 8) sin 8 aid EEL Sa De Ape epg ey S| sIn w sIn w w being the angle of ordination. It is clear that the equation of the line r is a-h y-k iy 0 Wes, Hence the results of this proposition must be stated in the follow- ing manner when the co-ordinates are oblique. PLANE CO-ORDINATE GEOMETRY. 175 (1) The condition-necessary in order that (AA) may be the middle point of the chord whose equation is a-h y-k m n is (Ah+ Bk + D)m+(Bk+ Ch + H)n = 0. (2) The equation of the diameter of the chords parallel to ds oe m n is (Am + Bn)x+(Cn+ Bm)y + Dm+ En= 0. -h y-k 6. —_ 2 point (hk), ten w Ah+ Bh+D m Ch+ Bh+ EF” The results (4), (5), (6), and (7) require no modification. be the equation of the tangent at the 413. All the propositions which follow Prop. cxx1, except Prop. cxxitl. are equally true for oblique co-ordinates as for rect- angular, provided we put m and » for cos 9 and sin 6, and 3 for tan 0. 414. Inthe case of Prop. cxxmt. the condition necessary, in order that the diameter be parallel to its chords, must be obtained from Art. 91. Let the equation of one of the chords, and the equation of the diameter, be Die Geo Line or nz — my = C, m n J silt = Yaa or naz-—my= C'. m n Then, by Art. 91, the condition necessary, in order that these two lines be perpendicular, is n(n +m cos w) + m' (m+ N COS w)=0....-- Wale but, by (2), Art. 412, we have m Cn + Bm nm Am+ Bn Hence (1) becomes (4m + Bn)(n +m cos w) = (Cn + Bm)(m +n cos ), 176 PLANE CO-ORDINATE GEOMETRY. which is the condition required, and agrees with the last result of Art. 382. It may be put im the forte m (A cos w~ BY" (A - CO ees Ccos w = 0, which gives two values of m, and therefore two sets of chords — whose diameters are at right angles to them. Various Problems. 4 415. ABC is a triangle, whose sides AB, AC, BC, always — pass through three given points (hk), (h’k'), (h'k’), respectively ; and the angular points A and B move along two given right — lines OAX, OBY, respectively : to find the locus of C. ; Taking OX and OY as co-ordinate axes, let the equations — AB, AC, BC be axe By = hoe JPL ee (1), ax + By =k ova ie woo kees iit See (2); art Sy e Lack Sau ance eee (3). Observing that, since (1) and (2) meet in the axis of z, and (1) ~ and (8) in the axis of y, (1) and (2) must give the same value of — z when y = 0, and (1) and (3) the same value of y whenz=0. _ Since (1), (2), (3) always pass through (24), (A'X’), (h’h’), — respectively, we have ah BR adil. 0. dee. ee ee (4), aha Oi ath i Ty wee (5), aA Dre. ee (6). And, if we eliminate a, 3, a’, 3’, between (2), (3), (4), (5), (6), the result will represent the locus of the intersection of (2) and _ (3), ee. the locus of C. Now, #' (2)-y (5), and h' (3) - 2 (6) give a(kz—-h'y)=k'-y, B(h'y - k'x) = h' - x. Hence, by (4), we have h(k' - ACh -y) , k (h' — zx) kiz — h'y Tay a which represents the locus required, which is therefore a line of © the second order. PLANE CO-ORDINATE GEOMETRY. 177 416. To find the locus of the intersection of two tangents drawn at the extremities of a chord of a conic section, supposing that the chord always touches another conic section. Take the axes of the latter conic section as co-ordinate axes, let its equation be hs 2 —=+ > a | MAS te tice cot ee at Wl and let the equation of the former conic section be Ax? + 2Bzy + Cy? + 2Dxe+2Fy+ F=0.... (2). Let (Ak) be the point of intersection of any two tangents of (2), then the equation of the chord joining the points of contact is (Ah+ Bh+D)x+(Ch+ Bh+ FE) y+ Dh+ Ek+ F=0...(3). Now if (8) be a tangent of (1), we have, by Art. 277, (Ah+ Bhi Df a? +(Ch+ Bh + EY 0 =(Dh+ Ek+ FY, which, being a relation between the co-ordinates, / and k, of the point of intersection of two tangents, drawn at the extremi- ties of any chord of (2) which touches (1), is the equation of the locus required. The locus is, therefore, a line of the second order. 417. The equations of four right lines being UE UO oO cog oie atte ciiiaiels eee ON OPAL Yt Coss lee pets ovals eae Gray byes Of Sk, aa) oa 2 cae a Guus EF eh Ey ome crete a beni gt aDs to shew that the general equation of a line of the second order, which passes through the points of intersection of (1) and (2), of (2) and (3), of (3) and (4), and of (4) and (1), is \ (ax+by +e) (az+Byt+y) +X (ae+by+c) (az+ Py +y7')=0...(5), \ and X’ being arbitrary constants. Let «, w', v, v' represent the first members of (1), (2), (3), (4), respectively ; then the equation uv + Nu'o' + pu’ + wu'o + vuu'+v'vv' =0.... (6), where XX’ ny’ vv’ are arbitrary constants, is evidently an equation of the second degree in z and y, and may be made to agree with the general equation of the second degree by giving proper N 178 PLANE CO-ORDINATE GEOMETRY. values to the six arbitrary constants. We may therefore regard (6) as the general equation of the second degree. Now if (6) be the equation of a curve which passes through the points of intersection of (1) and (2), of (2) and (8), of (8) and (4), and of (4) and (5), it is clear that, when we give z and y such values as make w and w’ zero, wv’ and v zero, v and v’' zero, or © and w zero, (6) must be satisfied ; which requires that p, py’, v, and v’ shall each be zero. Hence (6) becomes Awv + A'u'v' = 0; which coincides with (5). Q..D. 418. A conic section and a right line are given, and any quadrilateral is inscribed in the conic section ; to shew that, if three sides of the quadrilateral cut the given line in three given points, the fourth side will also cut it in a given point. Take the given line as axis of x, and any other line as axis of y, and let the equation of the conic section be Ax’ + 2Bry + Cy’ + 2Dxe+2Hy+ F=0......(1). Let the equations of the four sides of the quadrilateral be a2 +.by/+ 1 30,1415. « baneee see ia ne ie az+PBy+1=0.... Ra az+By+1=0.. . (5); then, by the preceding problem, the equation of cna conic section must be d (aa + by +1) (art By +1)4+XA(a'e+b'y4 1)(a' x+P'y+1)=0...(6); and this equation must agree with (1). We have, therefore, A =daa+X da, 2D=X(a+a)+X (+a), F=r4+X'; and from these equations, eliminating \ and X’, we may find a’ in terms of a,a@’,anda. Hence, if a, a’, anda be given, a’ is also given; but, when (2), (3) and (4) cut the axis of z in three given points, a, a’, and a are given; therefore a’ is given, and therefore (5) cuts the axis of z in a given point. Q.E.D. 419. In the preceding problem, if three sides of the quadri- lateral be parallel to three given lines, the fourth side will also be parallel to a given line. PLANE CO-ORDINATE GEOMETRY. 179 Let the equations of the sides be y+axr+b=0.... (2), Pile BPO.) 0% 2 (8); y+ar+P=0.... (4), Yer ae yr Oye) s hence the equation of the conic section must be \(y+ax+b)(y+ax+B)+r (y+ axe+0)(yt+axt P')=0...(6). Comparing this with (1), we find C=rX+N, 2B=X\(a+a)+rXN (a +a), A=daa+ Naa. Now, if (2), (3), and (4) be parallel to three given lines a, a’, and a are given; therefore, reasoning as before, a’ is also given, and therefore (5) is also parallel to a given line. 420. Band Bare given points, OAX a given line, and the right lines BP, B'P make given angles with the right lines BA, B'A respectively : to find the locus of P, supposing A to move along OX. Let OX be the axis of z, and a line at right angles to it through O the axis of y: let the co-ordinates of B and B' be, respectively, A& and h’k'; and let OA =a. ‘Then it is easy to see that the equations of B.A, B'A, BP, B'P must be, respectively, y- k=," @-#) pe pene (1), y — hi = —(@ ~ +) eee (2), Went eeqONE SION tem) ce Ree (3), Td aN GeV Ug Marcle cr greene (4), a, 3, and [3' being variable parameters. Let the given angles which (1) and (2) make respectively with (3) and (4) be tan™ m, and tan? m'; then we have k k(3 AEE ONE Coe 13 2 h-a P m/( + | k! : , aGy \ Siren Ce 1l+—— |}. hi -a P m( a) 180 PLANE CO-ORDINATE GEOMETRY. If we eliminate a from these equations, we obtain an equa- tion of the form 488 + BB + CB'+D=0..........(5), A, B, C, D, being constants: and, if we eliminate (3 and [3’ be- tween (3), (4), and (5), we evidently obtain the equation of the locus of the intersection of (8) and (4). Hence the required locus is represented by A (y-k) (y-k') +.B (y-k)(a-h') + C(y-k’) (a-h) +D (x-h) (x-h') = 0, an equation of the second degree. ‘The locus required is there- fore a line of the second order. END OF PART I. Metcalf &Palimer, Lithe, Cambridae Lael a yt e AP ie tt Ane Ry eae (38.) es (45 = o/7 P’ ZF Bee. = Pee af SS A Gay / ES A2, ; ef : (43 bis.) £ ¥ P 0 = Soe es : Bes | o Metcalfe 4 Palmer, Litho: Cambridge. Metcalfe & Poloner, Tithyg- Plate v- Hy cambridge. Metw dorks and Net Fdltions PUBLISHED BY J. DEIGHTON, CAMBRIDGE, BOOKSELLER TO H.R. H. THE CHANCELLOR, AND AGENT TO THE UNIVERSITY. MATHEMATICAL WORKS. 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