GEOMETRY OF CONICS. tamibititbe:. PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. THE GEOMETRY OF CONICS. PART I. BY C. TAYLOR, M.A. FELLOW 0F ST JOHN'S COLLEGE, CAMBRIDGE. CAMBRIDGE: DEIGHTON, BELL, AND CO. LONDON: BELL AND DALDY. 1872. [All Rights reserved.] PREFACE. THE work now in part published is the result of an attempt to reduce the chaos of Geometrical Conics to order. The subject having suffered not a little from desultory treatment, I have endeavoured to reconstruct it on a uniform plan, taking as a standard whereby to regulate the sequence of proofs the principle that Chord-properties should take precedence of Tangent-properties, the latter being deduced from the former rather than the former from the latter. This principle commends itself from more than one point of view. In proofs of chord-properties it is superfluous!to anticipate the comparatively complex notion of a tangent; while on the other hand the idea of a limit is the more clearly apprehended when approached in its natural order, and the properties of tangents are proved most convincingly when their intrinsic relation to the properties of chords is shewn. But, above all, this order refers the student consistently to first principles, thus rendering Geometry at once more thorough as a means of intellectual training, and more effective as an introduction to the powerful modern methods of Analysis. Simplicity is of course essential in an elementary work; but I have attempted at the same time to secure comprehensiveness by assigning their due importance to properties by which the Conic is characterized in the Higher Geometry of Curves-such as the quadratic relation between its coordinates and the rectilinearity of its diameters. As regards the practical working of this arrangement, it is scarcely too much to say that in the Parabola the student who lias thoroughly grasped Prop. II. has little more to learn. In the RectanT. b vi PREFACE. gular Hyperbola the corresponding theorem leads by easy stages to the solution of all difficulties; while in the Ellipse and the general Hyperbola a single construction* suffices for the demonstration of all the leading properties of conjugate diameters. Orthogonal Projection, already introduced under a new name, will be further discussed in the sequel. The Right Cone will be treated somewhat less inadequately than heretofore; but, despite the skilful advocacy of Mr Stuart Jackson, I am unable to acquiesce in the primary definition of Conics from the solid. No allusion has yet been made to the Conjugate Hyperbola, which may be viewed as a contrivance for giving a false definiteness to the student's conceptions, and perpetuating his illusion that the Hyperbola is a discontinuous curve. For the proof 1 of the chord-property that PN2 varies as AN. NA' my best thanks are due to Mr W. Allen Whitworth. The same proof was discovered independently by Mr Besant. In respect of the general tangent-property of Art. 108, I have pleasure in repeating my acknowledgments to Professor Adams. Mr Drew kindly places at my disposal his well-known proof that the tangent makes equal angles with the focal radii to its point of contact. In the Rectangular Hyperbola I have endeavoured to do justice to the investigations of Mr Wolstenholme, who has thrown much light upon that remarkable curve. I have profited by the advice and assistance of Mr Rawdon Levett of King Edward's School Birmingham, Secretary of the Association for the Improvement of Geometrical Teaching. It is needless to add that I am a debtor to Dr Salnon's inexliaustible works. April, 1872. * The trian-les pCN, dCR in Arts. 72, 212 differ only in relative position. t See page 38. Il Arts. 53, 202. CONTENTS. CHAPTER PA E I. CHORD-PIOPERTIES OF THE PARABOLA..... 1 II. TANGEKNT-PEOPIRTIES OF THE PARABOLA..... 12 III. CHORD-PROPERTIES OF THE ELLIPSE...... 24 IV. TANGENT-PROPERTIES OF THE ELLIPSE.....43 V. GHORD-PROPERTIES OF THE RECTANGULAR HYPERBOLA...61 VI. TANGENT-PROPERTIES OF THE RECTANGULAR HYPERBOLA.. 74 VII. CHORD-PROPERTIES OF THE HYPERBOL....78 VIII. TANGENT-PROPEITIES OF THE HYPERBOLA.....83 INTRODUCTION........... ix rTHE proofs in this work are so far independent that various propositions may be conveniently omitted in the first reading. The order of the Chapters may also be varied. The Rectangular Hyperbola may with advantage be studied very early. Of all Conics it is that which in its properties is most nearly identical with the circle; and the conception of its form is simplified by the use of the asymptotes as guiding lines. The figures for the general Hyperbola are so arranged that they miay be compared with the figures for the Ellipse. THE GEOMETRY OF CONICS. INTRODUCTION. THE name CONIC is applied to a family of. curves to which the circle belongs. It includes three varieties, the Ellipse, the Parabola, and the Hyperbola. THE ELLIPSE is the simplest in form. Its relation to the circle is shewn by the following construction, which will be found to lead to important results. From points in a circle let fall perpendiculars pN, dR,...to a fixed diameter AA'. If these perpendiculars be cut in a constant ratio thé points of section will lie on an ellipse. [73. By making the constant ratio of PIV to pN very small we may flatten tle ellipse indefinitely. And by making the ratio approximate to a ratio of equality we may make the ellipse as nearly circular as we please. A construction much simpler in practice results from using two circles, thus: Let AA', BB' be diameters at right angles of two concentric circles. If from the points in which a common radius cuts the circles parallels be drawn to BB', AA' respectively, their intersection will describe an ellipse. [Ex. 130. The following construction shews the relation of the ellipse to two points called the foci *. Let a loop of string, * The planets describe approximately ellipses having the sun in one focus. For this reason the first letter of Sol is used, as by Newton, to denote a focus. X INTRODUCTION. supposed inelastic, be passed over two pins at points S, S' (fig. p. 47), and let it be stretched into the form of a triangle, always in the same plane, by the point of a pencil or sharp instrument at P. Then as P moves in such a way as to keep the string stretched it describes an ellipse. It is evident that if S, S' be made to coincide the ellipse becomes circular, and if the distance SS' be made as great as possible, i. e. equal to half the length of the loop, the ellipse becomes flattened indefinitely. Example 98 gives another construction for thé ellipse. THE PARABOLA is the curve which would be described by a particle moving in a vacuum under the influence of gravity. A practical construction is given in Example 11. THE HYPERBOLA is most simply drawn by the analogous construction of Example 213. THE CONE. The three curves considered above were originally treated as plane sections of a Cone. Hence their old name Conic Sections. The cone and its sections may be shewn by means ot a wooden model. An ellipse may also be cut froi a cylinder or roller of circular transverse section. If the roller be cut obliquely, the section, supposed plane, will always be an ellipse. We may shew the sections optically by casting the shadow of a sphere or of a circular disc from a point of light upon a plane surface*. If the point of light be vertically under a point in the rim of the disc, the shadow thrown upon a vertical wall will be parabolic. These constructions and illustrations will suffice to give a practical acquaintance at the outset with the forms of Conics. We shall not, however, in our fundamental definition make ise of any of the constructions already given. * They may also be shewn very roughly by means of a light held at the small end of a conical lamp shade. INTRODUCT ION. xi LEMMAS. A. To prove geometrically that (a + b)2 - (a b)2 = 4ab. If four rectangles of sides a, b be fitted symmetrically about the square on a - b, the whole figure thus made up will` be the square on a + b. Therefore (a + b)2 = (a - b)2 + 4ab, or (a + )2 - (a -b) = 4ab. The same is proved in Euc. Ii. 8, but by an unsymmetrical construction which shews only a gnomon of equal area insteal of the four rectangles. B. From the ends of a straight line QQ' and from its middle point v (fig. p. 6) let parallels QM, Q'M', v be drawn to meet any other straight line. Draw parallels from Q', Q to M'M, and let them meet v O in Z', Z. Then in the triangles QvZ, Q'vZ', vZ= vZ', or QM- Ov= Ov-Q'M'. Therefore QM+ Q'M'i=2 Ov. If the figure be drawn as on p. 66, where O bisects Qq, it may be shewn in like manner that QM- qm = 20L. xii INTRODUCTION. If the parallels drawn as above be called the ordinates of the points from which they are drawn, we may enunciate the Lemma as follows: The ordinate qf the centre of a straight Une is equal to half the sum or difference of the ordinates of its ends. As a particular case of the above, if S be a point in the straight line itself, then SQ SQ'= 2v. For, SQ Sv QQ' Sv +SQ'; [Fig. p. 6. and similarly when S cuts Q Q' externally. C. In the figure of Art. 78, let PN be drawn perpendicular to SS'. Then by Eue. il. 12, 13, sP2 = CS2+ CP2 + 2 CS. CÎ, and S'P2 = CS'2 + CP - 2 CS'. CN. By addition, if C bisects SS', P+ S'P 2 = 2 CP2 +2CS2. D. In Art. 96, let a parallel from O to RS meet SP in M. Then SM: OR = SP: PR. And because the bisector of the angle PSQ is perpendicular to OM, and is nearer to SO than to SM; therefore SO is less than SM. Therefore SO: OR < SP: PR. DEFINITION. The term Equivalent will be used in this work to denote equal and similar or equal in all respects. CHAPTER I. CHORD-PROPERTIES 0F THE PARABOLA. DEF. A parabola is the curve described in a plane by a point which moves in such a way that its distance from a certain fixed point, called the Fiocus, is always equal to its perpendicular distance from a certain fixed straight line, called the Directrix. 1. With any straight Une MX as directrix, and any point S exterior to it asfocus, aparabola may be described. Fro any point M on th directrix draw a straight line From any point M1/ on the directrix draw a straight line at right angles to the directrix. On the straight line so drawn one point P belonging to the parabola can be found; for if the angle MSP be made equal to the known angle MSX, or SMP, then SP= PM, or P is a point on the curve. In this way any number ôf points P, P' belonging to the parabola may be found; and they all lie on the same side of the directrix with the focus. The straight line through the focus at right angles to the directrix is called the Axis. The point in which the axis meets the parabola is called the Vertex. Let the axis meet the directrix in X. The vertex A is the centre of the straight line SX, since from the definition the distance of A from S must be equal to its perpendicular distance AX from the directrix. T. 1 2 CHORD-PROPERTIES OF THE PARABOLA. 2. From the construction already given it appears that on every straight line PM perpendicular to the directrix, or parallel to the axis, one point and one only belonging to the curve can be found. In other words, if a straight line parallel to the axis cuts the parabola in a point P it cannot cut it again. Hence the parabola is an open curve; and it spreads out to infinity on both sides of the axis, since the point M, and therefore also P, may be taken as far as we please from the axis and on either side of it. The curve recedes at once from the directrix and the axis, since the angle PSX, or 2MSX, increases with MX. DEF. The perpendicular PN let fall upon the axis from any point P is said to be the Principal Ordinate, or briefly the Ordinate of P; and AN is said to be the Principal Abscissa, or briefly the Abscissa of P. Since NX = PM = the perpendicular distance of P from the directrix, therefore if P be a point on the parabola, SP = NX. It is sometimes convenient to express the definition in this form. 3. PROP. I. If PN, AN be the principal ordinate and abscissa of any point P on the parabola, then PN 2 = 4AS.AN. From the definition SP2 =NX2. Hence (ANA AS)2 + PN2= (AN+AS)2 [Euc. I. 47. = 4AS. AN+ (AN- AS). [Introd. Therefore PN2 = 4A S. AN. CHORD-PROPERTIES OF THE PARABOLA. 3 DEF. A straight line joining any two points on a curve is said to be a Chord. A chord which passes through the focus is said to be a Focal Chord. The focal chord perpendicular to the axis is called the Latus Rectum. 4. COR. The square of the focal ordinate or semi-latus rectum is equal to 4AS. AS. lence the latus rectum is equal to 4AS. Hence also, the ordinate is a mean proportional between the abscissa and the latus rectum. 5. Conversely, if P be such that PN2= 4AS. AN, where the length AS is gîven, or in other words, if PN' vary as AN, then will the locus of P be a parabola. The parabola may be very simply described with the help of Prop. I, To find successive points on the curve, we may proceed as follows. Measure off an abscissa AN of any length, for example, of length 3AS, Then by the proposition, PN = 4AS. 3AS. Therefore PN= 2 3AS. Now through the point N draw a straight line perpendicular to the axis, and on the line thus drawn take a point P at distance 2 v/3AS from the axis. Then P is a point on the parabola. Thus by measuring off abscissse of different lengths and using Prop. I. to find the lengths of the corresponding ordinates, we may determine successively any number of points on the parabola. In this construction the ordinates may be measured either upwards or downwards from the axis. Thus points on the curve are found in pairs as P, P' which have a common abscissa and equal ordinates. Hence the parabola is said to be symmetrical with respect to its axis, as may be seen likewise from Art. 1. EXAMPLES. 1. If the ordinates of a parabola be cut in a constant ratio the points of section will lie on a parabola. 2. Circles whose radii are in arithmetical progression touch a given straight line on the same side at a given point. If to each circle a tangent parallel to the given line be drawn, it will cut the circle next larger in points lying on a parabola. 1-2 4 CHORD-PROPERTIES OF THE PARABOLA. 6. PROP. II. The locus of the middle points of any system of parallel chords of a parabola is a straight Uine parallel to the axis. And the bisecting ine meets the directrix on the straight line through the focus perpendicular to the chords. 0 Y M __ Take QQ', any one of a system of parallel chords. Let the focal perpendicular on the chords meet QQ' in Y, and the directrix in O. Let fall perpendiculars QM, Q'M' on the directrix. Ther shall OM= OM'. For OM2 + QM2 0 Q2 [Euc. I. 47. = SQ2 + SO2 + 2 OS, SY. [Euc. II. 12. And QM2 = SQ2, from the definition. By subtraction OM2= SO- + 2 OS. SY. Similarly 06'12= S02 + 2 S. SY. Hence OM= OM', and the straight line through 0 parallel to the axis bisects Q Q' that is to say, it bisects every chord to which OS is perpendicular. 7. Otherwise, by Eue. i. 47: Since the squares of 0 Y, QY, and those of SY, Q Y are equal respectively to 0 Q2, and to SQ2, therefore 0 Y2 - _Y2 = Q2- SQ2= 0 Q2- QM2 [Def. = OM2 And OMJ'2 has the same value. CHORD-PROPERTIIES OF THE PARABOLA. 5 DEF. A straight line which bisects a system of parallel chords is called a Diameter. 8. In the parabola, all diameters are parallel to the axis. Conversely every straight line parallel to the axis is a diameter. It is evident that a diameter is determined when the centres of any two of its bisected chords are given. The axis may be called the Principal Diameter. 9. It is important that the student shoul. familiarize himself with Prop. I. It may be well to state it in converse forms thus: (i) If QQ' be any chord, and if the perpendicular upon it from the focus meet the directrix in 0, then if 0V be drawn to the centre of the chord it is parallel to the axis; or if it be drawn parallel to the axis it bisects the chord. (ii) If the diameter through the centre V of any chord meet the directrix in 0, and if OS meet the chord in Y, then 0 YV is a right angle. In the particular case of a focal chord bisected in v (fig. Prop. III.) z OSv =a right angle; and conversely if OSQ be a right angle, thé diameter through O bisects QQ'. EXAMPLES. 3. Using the method and notation of Art. 159, shew that for the parabola OL. QK=SX. qK. Apply this result to prove Prop. II. 4. If from the point O in which a diameter meets the directrix OZ be drawn parallel to the bisected chords and meeting the axis in Z, then OX = SX. XZ. 5. The circle on a chord of a parabola as diameter does not meet the directrix unless the chord passes through the focus. 6. Shew that z QOQ' = MYYM'. [6. 7. The perpendicular to a chord of a parabola from its middle point V meets the axis at a distance equal to SX from the foot of the ordinate of V. 6 CHORD-PROPERTIES OF THE PARABOLA, 10. PROP. III. To find the length of any focal chord. Take the Doint 0 on the directrix such that z OSQ = a right angle. Let a straight line through 0 parallel to the axis meet the curve in P and the chord in v. Then v is the middle point of QQ'*. [9. Let fall perpendiculars QM, Q'M' upon the directrix. Then SQ = QM, and SQ'= Q'M'. [Def. By addition, Q Q'= Q + Q'M' = 20v, since v is the centre of Q Q'. [Introd. But because P lies on the curve, SP = PO. Therefore the angles PSO, POS are equal, and their complements PSv, PvS are equal. Therefore Pv = PS = PO, or Ov = 2SP. Therefore Q Q' = 2 Ov = 4SP. 11. We shall now give more general definitions of the terms Ordinate and Abscissa. Let a diameter meet the curve in P, and meet QQ', any one of the chords which it bisects, in V. Then QV is the Ordinate and PVis the Abscissa of Q. More generally, if Q be any point whatever in the plane of the parabola, and Q V be measured parallel to the chords bisected by.PV, then Q V, PV are the ordinate and abscissa of Q referred to the diameter through P. The ordinates of all points on the curve referred to any fixed diameter are parallel. [6. * Or thus, without assuming Prop. II.: In the right-angled triangles MQO, SQO, which have MQ, QO=SQ, QO, the side OM=OS=OM', similarly. Hence Qv = Q'v. CHORD-PROPERTIES OF THE PARABOLA. T 12. PROP. IV. If PV be the abscissa of any point Q on the parabola, then QV2 = 4SP. PV. Let SY meet a chord Q ' perpendicularly in Y, and meet the directrix in 0. Draw a parallel to the axis through O meeting the curve, Q' Q, and the parallel focal chord, in P, V, v. Let fall perpendiculars QD, QM on 0V and the directrix. Then OY: OV= SY: vV, by parallels. Hence OY- SY2: OV'-vV2 = OY: O 0F2, = QD2 QV 2 by similar right-angled triangles O YV, QD KV But QD2 = 012 = 0 Y2 _ 2, as in Art. 7. Therefore QV 2 = O V2- VV2. Also Pv = SP= PO, as in Art. 10. Therefore OV= PV+ SP, and vV= PV- SP. Therefore OFV vV2 = 4SP.PV. [Introd. Hence Q V2 = 4SP. PV, from above. And, Q' V' having the same value, Q V is a semi-chord or ordinate, 8 CHORD-PROPERTIES OF THE PARABOLA. 13. COR. The ordinate QV is a mean proportional between the abscissa P V and the parallel focal chord. [10. 14. PROP. V. The rectangles contained by the segments of any two intersecting chords are to one another as the lengths of the parallel focal chords. P@\ Let the diameters through O the intersection of any two chords QQ', RR', and through V the middle point of Q', meet the curve in q, P respectively. Let Pv be the abscissa of q. Then QO. OQ' = QV- 0V2 [Euc. iI. 5, Cor. = Q V2- qv2 = 4SP. PV- 4SP. Pv. [12. But P V-Pv is equal to v V or q O. Therefore Q O. O Q' = 48SP. qO. Similarly, if P' be the point in which the diameter bisecting BR' meets the curve, RO. OR'=4SP. qO. Hence Q O. Q': R O. OR' = 4SP: 4SP'. 15. COR. The ratio of these rectangles is independent of the position of O*. It is the same for every pair of chords parallel to QQ', RR' respectively, since for all such chords P and P' remain unaltered. [6. * This point may be either external to the parabola or internal. EXAMPLES. 9 8. A point within a parabola is nearer to the focus than to the directrix. 9. All parabolas are similar curves. Prove this by deducing from the bisection of the angle PSX by SM that the parallel focal radii of similarly situated parabolas are proportional. [1. 10. If the focal radii of a parabola be cut in a constant ratio the points of section will lie on a parabola. 11. Prove the following construction. Take any ordinate YP, and draw PM parallel and equal to NSA. Divide NP into any number of equal parts, and through the points of section draw parallels p,,, p,,...to the axis. Divide MP into the same number of equal parts in points 1, 2, 3,... Then the lines p, p2, p3,...meet A, A2, A3,... respectively on the parabola. 12. Shew that z QOQ' = a right angle = MSM'. [10. 13. The semi-latus rectum is a mean proportional between the principal ordinates of the ends of a focal chord. And if AM, AM' be the corresponding abscissoe, then AM. AM' = AS2. 14. The circle on a focal chord touches the directrix. 15. Shew that S0 = SX. Ov= 4AS. SP. [12. 16. Also that QD = 4AS. PV. [12. 17. Deduce from Ex. 9, that the ratio of the rectangles in Art. 15 is the same for all similarly situated parabolas. 18. A circle can be described touching any two diameters of a parabola and the focal radii to their extremities. 19. The locus of the centre V of a focal chord is another parabola. Prove this by shewing that if VL be the ordinate of V, then VL= 2AS. AL. [Ex. 4. 20. A chord QQ' is cut in 0 by a diameter which meets the curve in P. Shew that if R be a point on the curve whose abscissa is PO, and PV, PV' be the abscissoe of Q, Q', then QVy-Q'V' r: QV OR2=QV+ Q'V': QV. [12. Deduce that OR is a mean proportional between Q, Q'V'. 10 CHORD-PROPERTIES OF THE PARABOLA. 21. Shew also that PV. PV'= PO'. 22. Shew how to place in a parabola a focal chord of given length. 23. A parabola being given find its axis and focus. 24. If PQ be a chord which subtends a right angle at A, and AN, AM be the principal abscissoe of P, Q, then PQ passes through a fixed point, and AN. AMi =PN. P = 16AS2. 25. If PQ be a focal chord, AP, AQ meet the latus rectum at distances from S equal to the ordinates Q, P. [Ex. 13. 26. If the diameter at P meets the latus rectum in L, the intersection of AL, XP is on the curve. 27. A point on a parabola being given, then if the focus be given the envelope of the directrix is a circle; or if the directrix be given the locus of the focus is a circle. 28. Given the directrix of a parabola and two points on the curve; shew that two positions of the focus can be determined. 29. Given the focus of a parabola and two points on the curve; shew that two positions of the directrix may be found. 30. If two parabolas have a common focus, their common chord passes through the intersection of their directrices and bisects the angle between them. 31. If two parabolas have a common directrix, their common chord bisects the straight line joining their foci at right angles. 32. The straight lines which join the ends of a focal chord to the vertex meet the directrix on the diameters through the ends of the chord. 33. Find the locus of the centre of a circle which passes through a given point and touches a given straight line. 34. A point moves in such a way that its perpendicular distance from a fixed straight line is less by a constant quantity than its distance from a fixed point. Shew that the point describes a parabola. 35. Find the locus of the centre of a circle which touches a given circle and a given straight line. EXAMPLES. l1 36. Circles being described on the segments of a focal chord as diameters, the straight line joining their centres subtends right angles at the intersections of their common tangents. 37. The perpendicular from Q to a chord AQ meets the axis at a distance equal to the latus rectum from the foot of the ordinate of Q. 38. The straight lines joining any point on a parabola to the ends of a focal chord intercept on the directrix a length which subtends a right angle at the focus. 39. Parallel chords being cut by any diameter, the rectangles contained by their segments are as the abscissoe of the points of section. 40. Equal focal chords of a parabola are equally inclined to the axis. Hence shew that the common chords of a circle and a parabola are equally inclined to the axis. [14. 41.,A circle cuts a parabola in three points, whether on the same side of the axis or on different sides. Prove that it cuts the curve in one other point and one only, and shew how to find this point. 42. If a circle cuts a parabola in points 1, 2, 3 above the axis and in a point 4 below it, the difference of the ordinates of 1, 3 is to the difference of their abscissoe as the sum of the ordinates of 2, 4 to the difference of their abscissoe. Deduce that the ordinate of 4 is equal to the sum of the ordinates of 1, 2, 3. 43. If 1, 2 and 3, 4 lie on opposite sides of the axis the sum of the ordinates of 1, 2 is equal to the sum of the ordinates of 3, 4. 44. On a chord through a fixed point O there is taken a mean proportional OM to the segments of the chord. Shew that the locus of M is a diameter. ( 12 ) CHAPTER II. TANGENT-PROPERTIES OF THE PARABOLA. 16. IF the extremities of a chord be made to approach one another continuously the chord tends to assume a certain limiting position; and ir this limiting position, viz. when its extremities have become coincident, the chord produced indefinitely has become a Tangent. Thus in fig. p. 14 suppose Q to move along the curve up to Q. Then ultimately QQ' becomes the tangent at Q. There are various ways in which a chord may be made to assume a position of tangency. It is often convenient to use the following. 17. In fig. p. 14 Qq is a chord bisected by the diameter TV. Suppose this chord to move parallel to itself into a new position Q q'. It is still bisected by TV, say in V'. [6. Suppose it now to move continuously in the same way until its middle point reaches the end of the diameter TV; then each of the portions Q V, q V, which have been diminishing continuously, will have vanished, and the two points in which the chord met the curve will have been brought into coincidence; that is to say the chord will have become a tangent, viz. at the end P of the diameter which bisects Qq. Hence The tangent at the extremity of any diameter is parallel to the ordinates of that diameter. In particular The tangent at the vertex is parallel to the principal ordinates, or perpendicular to the axis. TANGENT-PROPERTIES OF THE PARABOLA. 13 18. To take another example:-In Art. 14 we have shewn that the rectangles contained by the segments of any two intersecting chords are to one another as the lengths of the parallel focal chords. This holds whether the point of intersection be internal or external to the curve. Let it be external as T in fig. p. 14. Then TQ. TQ' is to Tq. Tq' in the ratio of the focal chords to which QQ', qq' are parallel. Let QQ' move parallel to itself until it becomes the tangent at some point Q1. Then TQ. TQ' becomes TQ12. In like manner let qq' become the tangent at some point q1. Then Tq. Tq' becomes Tq12. Therefore the squares of any two tangents TQ1, Tq1 are to one another in the ratio of the focal chords to which they are parallel. 19. Again (9), if a diameter meet the directrix in O, then every chord QQ' bisected by that diameter is perpendicular to OS. Therefore in the limit the tangent at P (fig. p. 7) the extremity of the diameter OV is perpendicular to OS, or, in fig. p. 16, to MS. And conversely, if the focal perpendicular' on the tangent at P meets the directrix in M, then PM is a diameter, and is therefore parallel to the axis or perpendicular to the directrix. 20. We might also, after the method of Euclid, regard a tangent as a straight line which meets a conic and being produced does not cut it, or which, except at the point of contact, lies wholly on the convex or outer side of the conic. We shall briefly indicate the method to be pursued when the Euclid definition is adopted. From any point P on the parabola (fig. p. 16) let fall perpendiculars PM, PY on the directrix and SM respectively. Then will PY be the tangent at P. For since evidently Y is the centre of SM, therefore if t be any point other than P on PY, then St= TM. Therefore St is greater than the perpendicular distance of t front the directrix. Hence it may be shewn that every point t on PY lies on the convex side of the parabola, or PY is the tangent at P. [51. 14 TANGENT-PROPERTIES OF THE PARABOLA. 21. PROP. VI. Tangents at the extremities of any chord intersect on the diameter which bisects the chord. ' V Let Q, Q' and q, q' be adjacent extremities of any two parallel chords, and let Q Q', q'meet in T, then shall T lie on the diameter which bisects the chords. For let TV, drawn to the centre of Qq, cut Q'q' in V'. Then by parallels, Q'': q'V'=QV: qV. But Q V= q V. Therefore Q' V' = q '. Hence TV, since it bisects both Qq and Q'q', is the diameter which bisects chords parallel to Qq. [8. Now as Q'q' moves parallel to itself up to Qq, the point of intersection T always lies on the diameter through V. And this being true always, is true in the limit, viz. when QQ', qq' become the tangents at Q, q. Hence the tangents at the'ends of the chord Qq meet on the diameter which bisects the chord. 22. Conversely, if the tangents at P, Q meet in R, the straight line drawn through R to bisect the chord of contact P Q is a diameter. TANGENT-PROPERTIES OF THE PARABOLA. 15 23. PROP. VII. If PV be the abscissa of any point Q on the parabola, and if the tangent at Q meet the diameter PV in T, then PV = PT.,1 Let the tangent at P, which is parallel to the ordinate Q V, meet QT in R. Complete the parallelogram QRPO, viz. by drawing PO parallel to RQ. Then the diagonal RO bisects the diagonal P Q. That is to say, R O bisects the chord of contact of the tangents RP, R Q. Therefore RO is a diameter (22), and is parallel to any other diameter, as PV, since all diameters are parallel to the axis. Hence by parallels, PV=RO=PT. 24. The following particular case is to be noticed. y t p n T principal rinae of P, then A = If PN be the principal ordinate of P, then AN= AT. 16 TANGENT-PROPERTIES OF THE PARABOLA. 25. PROP. VIII. The tangent at P bisects the angle between SP and the diameter through P. Draw PM perpendicular to the directrix, and let SMmeet the tangent at P in Y. Then in the triangles SYP, MYP, since SP, PY= MP, P Y, each to each, and the angles at Y are right angles, [19. therefore z SP Y= MP Y. 26. COR. If the tangent meet the directrix in R, it readily follows that z RSP = RMP = a right angle. But this is best proved independently as in Art. 105. 27. COR. If the tangent meet the axis in T, then by parallels and by the proposition, z STP = MPT= SPT. 28. PROP. IX. If from any point T on the tangent at P perpendiculars TL, TN be let fall on SP and the directrix, then SL =TN. If the tangent meet the directrix in R the angle RSP is a right angle. [26.. TANGENT-PROPERTIES OF THE PARABOLA. 17 Therefore TL, RS are parallel. So too are TN, PM. Hence SL: SP= TR PR [Euc. vi. 2. = TNV: PM. But SP = PM. Therefore SL = TN. 29. PROP. X. If the tangents at P, Q meet in T, TP, TQ subtend equal angles at S. On SP, SQ and the directrix let fall perpendiculars TL, TM, TN. Now because T is a point on the tangent at P, therefore SL = 'N. [28. And because T is a point on the tangent at Q, therefore SM= T.V. [28. Hence SL, SM are equal. Therefore in the right-angled triangles STL, STM the angles at S are equal; that is to say, TP, TQ subtend equal angles at S. EXAMPLES. 45. The portion of any tangent intercepted by the tangents at the ends of a parallel chord is bisected at the point of contact. 46. Apply this result to determine the length of a focal chord. 47. Any length measured on a tangent subtends equal angles at the focus and at the point in which the diameter through the point of contact meets the directrix. T. 2 18 TANGENT-PROPERTIES OF THE PARABOLA. 30. PROP. XI. The acute angle between any two tangents is equal to half the angle which their chord of contact subtends at thefocus. %P O Let the tangents at P, Q intersect in T and meet the axis in p, q. Take any point 0 in pS produced. Then by Euc.. 32, zPSO= SPp + SpP= 2SpP. [27. Similarly, z QSO = = 2SqQ. Hence: (i) If P, Q lie on opposite sides of the axis, then by addition, z PSQ = 2 (SpP+ SqQ) = 2 (SpP+pqT) = 2PTQ. [Euc. 1. 32. (ii) If P, Q lie on the same side of the axis*, then by subtraction, z PSQ = 2 (SQ - SpP) = 2p Tq. Therefore in each case the acute angle between the tangents is equal to half the angle PSQ. 31. PROP. XII. If the tangents at P, Q meet in T, the triangles SPT, STQ are similar. * Use the figure on p. 17, supplying the letters p, q, O. TANGENT-PROPERTIES OF THE PARABOLA. 19 If P, Q lie on opposite sides of the vertex, then in the preceding figure produce TS to t. Then z PSt = PSQ [29. =PTQ. [30. Hence by Euc. I. 32, z STP+ SPT = STQ + STP. Therefore z SPT= STQ. And zTSP= TSQ. [29. Therefore the triangles SPT, STQ are similar. 32. PROP. XIII. If SY be the focalperpendicular on the tangent at P, then Y lies on the tangent at A, and SY2 = SA. SP. Let SY meet the directrix in M. Then PilI is a diameter (19), and is therefore perpendicular to the directrix. Hence in the right-angled triangles SPY, MPY, SP = PM. Also PY is common. Therefore SY= MY. Hence A Y, since it bisects both SM and SX, is parallel to MX, and is therefore the tangent at A. [17. And since, from what precedes, the triangles SPY, MPY are similar, therefore z PSY = PMY= YSA, by parallels. Hence by similar right-angled triangles SPY, SYA, Y: SP= SA: SY, or SY2= SA.SP. 2-2 20 TANGENT-PROPERTIES OP THE PARABOLA. 33. COR. Any two tangents TP, TP' are as SY, SY' the focal perpendiculars upon them. For TP2: TP'= SP: SP' [18. = 82: SY'2. 34. DEF. The Normal at any point of a curve is the straight line drawn through that point at right angles to the tangent. The Subtangent is the portion of the axis intercepted between the tangent and the principal ordinate. The Subnormal is the portion of the axis intercepted between the normal and the principal ordinate. 35. PROP. XIV. The subtangent is equal to the principal abscissa; and the subnormal is equal to half the latus rectum. (i) Prove as in Prop. viI. that AN= A T. Hence the subtangent NT= 2AN. (ii) Let the normal at P meet the axis in G; and let the diameter through P meet the directrix in M. Then PG, being at right angles to the tangent, is parallel to MS. [19. -T - -A S -- - Hence the triangles PNG, MXS are similar; and, being also of equal altitude, they are equal in all respects. Therefore the subnormal NG = SX = 2A S. 36. Or thus: Because TPG is a right angle, therefore PN2 = NG. NT = NG. 2AN, from above. But PN2= 2AS. 2AN. [3. Therefore the subnormal NG = 2AS. TANGENT-PROPERTIES OF THE PARABOLA. 21 37. It should be noticed that SG= SP= ST. [27. 38. In the figure on p. 6, since OS is at right angles to QQ', therefore OQ, OQ' are the tangents at, Q'. [26. And, since the right-angled triangles OSQ, OMQ have their sides SQ, QM equal, and Q O common, therefore z QOS= QOM = MOS. Similarly z Q'OS= M'OS. By addition z Q O = a right angle. Hence in the parabola tangents at the ends of a focal chord meet at right angles on the directrix; and conversely, the directrix is the locus of intersection of tangents at right angles. 39. It will be seen that some of the definitions in Chapters i. iI. are applicable to all conics; as, for example, of Axis, Vertex, Ordinate, Latus Rectum, Diameter, Tangent, Normal, Subtangent, Subnormal. So too is Art. 21. EXAMPLES. 48. Shew how to draw a tangent to a parabola from a point on the tangent at the vertex. 49. If a leaf of a book be folded so that one corner moves along an opposite side the direction of the crease touches a parabola. 50. Shew how to draw tangents to a parabola from any given external point. 51. If PQ be the chord of contact of tangents from 0, then SO0 = SP. SQ. 52. The tangent at Q meets the tangent at P in R and the diameter through P in T. If PT meets the directrix in M, shew that the triangles MPR, RPT are similar, so that PR' = PM. PT. 53. Hence shew that, P V being the abscissa of Q, QV = 4SP. P V. 54. Any two tangents TP, TQ make equal angles with TS and the diameter through T respectively. 22 TANGENT-PROPERTIES OF THE PARABOLA. 55. If from any point T on a fixed tangent a second tangent TP be drawn, the angle STP will be constant. 56. The vertices of any circumscribed triangle are concyclic with the focus. 57. PQR being a circumscribed triangle, the perpendiculars from P, Q, R to SP, SQ, SR cointersect. 58. If the tangents at P, Q meet in T, and if C be the centre of the circle TPQ, then CST is a right angle. 59. The normal at P bisects the angle between SP and the diameter through P. 60. Two parabolas which have a common focus and their axes in opposite directions intersect at right angles. 61. The perpendicular drawn to a normal from the point in which it meets the axis envelopes an equal parabola. 62. Normals at the extremities of a focal chord intersect on the diameter which bisects the chord. 63. If PQ be a focal chord, and R the foot of the perpendicular upon it from the intersection of the normals at P, Q, then SP = QR. 64. The portion of any tangent intercepted by the tangents at fixed points P, Q subtends a constant angle at S. The angle subtended is a right angle when PQ passes through S. 65. If a circle through S touches the parabola in P, Q, then SP is equal to the latus rectum. 66. The normal at any point is equal to twice the focal perpendicular upon the tangent, and is also a mean proportional between the focal distance of that point and the latus rectum. 67. The squares of the normals at the ends of a focal chord are together equal to the square of twice the normal perpendicular to the chord. 68. The locus of the vertex of a parabola which has a given focus and touches a given straight line is a circle. 69. The circle on a focal radius touches the tangent at A. 70. The tangent and normal at any point are bisected by the focal perpendiculars upon them; and the straight line joining the feet of the perpendiculars is parallel to the axis. EXAMPLES. 23 71. The diameter through one end of a focal chord bisects the chord normal at the other. 72. The locus of the foot of the focal perpendicular on the normal is a parabola. 73. If the tangents at P, Q intersect in R, the circle through P touching QR in R passes through S. 74. If from the foot of the normal at P a perpendicular PK be drawn to SP, then PK = 2AS. 75. Determine the position of P so that the triangle SPG may be equilateral. 76. The tangent at any point meets the directrix and the latus rectum in points equidistant from the focus. 77. If QQ' be the focal chord perpendicular to the normal at P, then PG2 = SQ. SQ'. 78. The triangle bounded by three tangents to a parabola is equal to half the triangle whose vertices are at the points of contact. 79. If two parabolas be described each touching two sides of a given equilateral triangle at the points in which it meets the third side, prove that they have a common focus and that the tangent to either of them at their point of intersection is parallel to the axis of the other. 80. Two equal parabolas have the same axis and directrix. From a point on one of them tangents are drawn to the other. Shew that the perpendicular from that point to the chord of contact is bisected by the axis. 81. Supposing the triangle 123 in Ex. 42 to become evanescent, shew that in the limit the common chord of the circle and the parabola is equal to four times their common tangent measured from the curve to the axis. 82. A diameter meeting a chord and the tangent at an end of it is cut by the curve in the ratio in which it cuts the chord. 83. Draw a chord which shall be cut in a given ratio by a given diameter. 84. A parabola being inscribed in a triangle its directrix passes through the orthocentre. ( 24 ) CHAPTER III. CHORD-PROPERTIES OF THE ELLIPSE. 40. DEF. A Conic is the curve described in a plane by a point which moves in such a way that its distance from a certain fixed point, called the Focus, is in a constant ratio to its perpendicular distance from a certain fixed straight line, called the Directrix. This constant ratio is called the Eccentricity. A Conic is called an Ellipse, a Parabola, or a Hyperbola, according as its eccentricity is less than, equal to, or greater than unity. 41. Some properties can be proved as simply for all conics at once as for the ellipse separately. We shall accordingly, in enunciating some of the propositions in Chapters III. and iv., use or imply the general term Conie instead of Ellipse. Articles which apply to all conics will be distinguished by the mark 1T. 42. Let S be the focus of an ellipse, and X the point in which the axis meets the directrix. Divide SX in A so that SA may be to AX as the eccentricity. Then A is a vertex. Since the eccentricity is less than unity it is evident that there is a second vertex A' in XS produced, such that SA' is to A'X as the eccentricity. 43. The ellipse lies wholly on the same side of the directrix with the focus. For imagine two of its points 0, P to lie on opposite sides. Let OP cut the directrix in R. Then, CHORD-PROPERTIES O' THE ELLIPSE. 25 by the definition if OP be at right angles to the directrix, and otherwise afortiori, SO < OR, and SP < PR. O.R xl' mA\ + 4\C' a' By addition, SO + SP < OP, which is impossible. Hence the ellipse lies wholly on one side of the directrix, viz., that on which are the points A, A' already determined. ~ 44. An extension of the definition of a conic. From any point P on a conic draw PM perpendicular to the directrix, and PR meeting the directrix at any constant angle. Then PM: PR is a constant ratio. But by definition SP: PM is constant. Therefore SP: PR is constant. Hence a conic might have been defined as the locus of a point P whose distance SP from the focus is in a constant ratio to its distance PR from the directrix measured parallel to any fixed straight line which meets the directrix. When this fixed straight line meets the directrix perpendicularly we come back to the original definition. ~145. As a particular case of the above let P, Q be two points on a conic*, and let the straight line joining them meet the directrix in R. Let fall perpendiculars PM, QN on the directrix. Then SP: SQ = PM: QN [Def. =PR: QR, by parallels, and conversely if this relation holds and P be on the curve, then Q will be on the curve. * This includes the case of a focal chord. 26 CHORD-PROPERTIES OF THE ELLIPSE. 46. For two chords of an ellipse through P and A, A' respectively which meet the directrix in Z, Z', z7 SA: SP = AX: PM [Def. = AZ: PZ, by parallels; and SA': SP = A'Z': PZ', similarly. ir47. From Art. 45, when one point P on the curve is given a second Q may be determined by drawing PR to any point R on the directrix, and then drawing SQ making the same angle as PS with SR. For SQ will meet RP in a point Q such that SP: SQ = PR: QR. [Euc. vI. A. From this construction it appears that a straight line which meets a conic in a point P will in general meet it in one other point Q, and that no straight line can meet a conic in more points than two. Hence Conics are called curves of tle second degree. 48. Starting from the vertex A we may determine any number of points as P on the ellipse, by drawing AZ to any point Z on the directrix, and then making the angle ZSp equal to the known angle XSZ. The line pS thus drawn meets ZA on the ellipse. [47. Or we miglit have determined points as P by starting from the vertex A' and taking the angle Z'SP equal to XSZ', where Z' is any assumed point on the directrix. CHORD-PROPERTIES OF THE ELLIPSE. 27 49. The point P determined by the second construction will not in general be the same point as that determined by the first. But it will be the same if Z, Z' be suitably chosen. To find the necessary relation between Z, Z' suppose them to be the points in which PA, A'P, drawn to the same point P on the curve, meet the directrix. Then z XSZ = supplement of XSP, [48. and z XSZ'= XSP. 2 By addition, z ZSZ' = a right angle. Hence in Art. 48 if the angles XSZ, XSZ', measured on opposite sides of the axis, be complementary, the two constructions give the same point P on the curve. Hence the simpler construction, Join AZ, A'Z' intersecting in P. 50. The point Z may be any point on the directrix. Suppose Z to start from an infinite distance below the axis, and to approach X. When Zis at infinity Z' must be at X; as Zapproaches the axis Z' recedes from it; when Z is at X, then Z' is at infinity. After this the points Z, Z' change sides of the axis, and each of the lengths ZX, Z'X passes continuously through the values which the other had before. Hence we infer that the ellipse is symmetrical with respect to its axis. And since the angle XAP, being greater than the interior angle AXZ, is obtuse, and XA'P is acute, therefore the ellipse is a closed curve lying wholly between the perpendiculars to the axis through the vertices. EXAMPLES. 85. Shew that Z'Ap, ZpA' are straight lines. 86. The straight lines joining a vertex to the ends of a focal chord intercept on the directrix a length which subtends a right angle at the focus. 87. The parallels to the axis through P, p cut ZS, Z'S in points Q, q such that Qq meets Pp on the directrix, and cuts the axis at a distance equal to the semi-latus rectum from S. 28 CHORD-PROPERTIES OF THE ELLIPSE. 1 51. PROP. T. The distance of any point P outside a conic from the focus S is to its perpendicular distance from the directrix in a ratio greater than the eccentricity. 's ~ P Let SP cut the conic in p, from which point let fall a perpendicular pm on the directrix. The ratio Sp: pm is equal to the eccentricity. Let pm eut SM in n. Then by parallels, SP: PM= Sp: pn > Sp pm. As the figure is drawn p falls between S and P. When P lies beyond the directrix, p is to be taken in PS produced. 1~52. When P is within the conic it may be shewn in like manner that SP: PM< Sp: pm, where p is to be taken in SP produced*. This proposition illustrates a remark made above; for it is immaterial whether the eccentricity Sp: pm be less than unity or wholly unrestricted. In other words, the theorem is proved as simply for all conics at once as it could be for the ellipse separately. EXAMPLES. 88. If an ellipse, a parabola, and a hyperbola have the same focus and directrix, the ellipse lies wholly within the parabola, and the parabola wholly within the hyperbola. 89. Conics having the same focus and directrix do not meet. * Except when P is within the further branch of a hyperbola. CHORD-PROPERTIES OF THE ELLIPSE. 29 53. PROP. II. The square of the principal ordinate PN of any point P on an ellipse whose vertices are A, A', varies as AN. NA'. AfA Let A'P, PA meet the directrix in Z', Z Then PN: AN =ZX: AX, and PN: A'N=Z'X: A'X. Compounding, PN2: AN. A'N= ZX. Z'X: AX. A'X = SX2:AX.A'X, since the angle ZSZ' is a right angle. [49. Therefore PN2: AN. A'N is a constant ratio. DEF. The middle point C of AA' is called the Centre* of the ellipse. The central chord BCB' perpendicular to the axis is called the Minor Axis, and AA' itself is also called the Major Axis. CN is called the Principal Abscissa of P. EXAMPLES. 90. Shew conversely that if PN2 varies as AN. NA', then ZX. XZ' is constant. 91. Determine the value of this constant by making N coincide with S. 92. In Art. 44, if SD be drawn parallel to PR to meet the directrix, SP: PR=L: SD. [67. * The ellipse and the hyperbola are called Central Conics. 30 CHORD-PROPERTIES OF THE ELLIPSE. ON THE SYMMETRY OF THE ELLIPSE. This result leads to a simple method of determining the form of the ellipse. 54. Let P3N coincide with BC. Then the constant ratio PN2: AN. NA' assumes the form CB2: CA2. Hence always V2 2: AN..NA' = CB2: CA2, or PN'~2:A CA-C2 = CB: CA2. When the length CN is known the length PN is known. And since P.V may be measured either upwards or downwards from thé axis, therefore, as we have already seen, the ellipse is divided symmetrically by its axis. When the length PNis known the length CNis known. And since CN may be measured either to the right or to the left from C the ellipse is divided symmetrically by its minor axis. 55. From any point P let fall a perpendicular Pn on BB', and produce it to P' so that P'n may be equal to Pn. If now, with some writers, we call P' the Reflexion ofP with respect to Be', then we may say that BB' divides the figure into two parts such that one is the reflexion of the other. The ellipse must therefore have a second focus S' the reflexion of S, and a second directrix the reflexion of the first. The second focus and directrix have the same properties as CHORD-PROPERTIES OF THE ELLIPSE. 31 the first, and the curve may be described equally well from either of the relations SP: NX =SA: AX, S'P: NX'= S'A' A'X' =SA: AX. T56. It is hence evident that equal focal radii, as SP, S'P', Sp, S'p' and equal focal chords are such as are equally inclined to the axis. 57. PROP. III. The sum of the focal radii to any point on the ellipse is constant. For SP: NX= SA: AX-= 'P: NX'. [55. Hence SP +S'P: NX+NX'= SA: AX; or SP + S'P: 2CX = SA: AX. Therefore SP+ S'P is constant. Let P coincide with A. Then the constant value of SP + S'P assumes the form SA + S'A, that is AA' or 2CA. Hence always SP + S'P= AA' = 2 CA. 58. Take any point Q outside the ellipse. On the arc intercepted by SQ, S'Q take any point P. Then SQ + S'Q is greater than SP+ S'P, or AA'. [Euc. I. 21. For an internal point, SQ + S'Q is less than AA'. The following results should be noticed: 59. When P coincides with B, SB + S'B= 2 CA. Hence SB= S'B= CA. 60. Hence CS2 = SB2 - CB2 = CA2- CB2. 61. Also AS. SA'= A2- CS2= CB2. 62. Again, SB: CX= SA: AX. [Def. Therefore CA: CX= SA: AX. 63. Hence CA-SA: X-AX= = CA: CX; or ÇS: A = CA: CX = SA: AX. 32 CHORD-PROPERTIES OF THE ELLIPSE. 64. Hence also CS: CX= CS2: CA2, and CS. CX = CA2. 65. Therefore CS. CX - CS = CA2 - C2, or CS. X = CB2. 66. The definition may be expressed in the form SP:PM= CS: CA, [63. or SP: NX = S: CA. 67. If L denote the focal ordinate or semi-latus rectum, then, writing S for N in Art. 54, L: A S. SA'= CB2: CA2, or L2: CB2 = CB': CA2. [61. Therefore CB is a mean proportional to CA and L. Or thus: L: SX= C: CA,- [66. therefore L. CA = CS. SX= CB2. [65. 68. PROP. IV. The locus of the middle points of any system ofparallel chords is a straight lne passing through the centre of the ellipse. ^Z; DR Let Qq be any one of the system of chords parallel to the radius CD; and let QM, qm be the ordinates of Q, q. Then QM2: CA2 - CM' = CB2: CA', [54. and qm2: CA2 - Cm2 = CB2: CA2. Hence QM2 - qm2: Cm' - CM2 = CB2: CA'. Bisect Qq in O. Let OL, DR be the ordinates of O, D. Draw qK parallel to the axis to meet QM. CHORD-PROPERTIES OF THE ELLIPSE.,:3 Then* QM - qm: Cm - CM = QK: qK =DR: CR, by similar triangles QqK, D CR. Also QM+ qm: Cm + CM= 2 OL:2 CL. [Introd. Hencet 2 OL. DR: 2 CL. CR = CB2: CA2, fiom above. Therefore, DR and CR being constant, the ratio OL: CL is constant, or the locus of O is a straight line through C. 69. Hence all diameters pass through the centre. It is evident that equal diameters are such as are equally inclined to the axis. Thus in the figure on p. 30, the diameters Pp', P'p are equal and equally inclined to the axis. 70. Produce CO to meet the ellipse in P. It follows at once from what is proved in Art. 68, that, PN being the ordinate of P, PN. DR: CN. CR = CB2: CA2. The symmetry of fhis relation shews that if CP bisects chords parallel to CD, then CD bisects chords parallel to CP. DEF. Two diameters are Conjugate when each bisects chords parallel to the other. Thus the axes are conjugate. More generally any two chords parallel to conjugate diameters may be called Conjugate Chords. Supplemental Chords are such as join the ends of any diameter to some point on the curve. 71. To shew that Supplemental Chords are Conjugate. In the figure of Art. 78 let SS' be a diameter of an ellipse; P any point on the curve; C the centre. Bisect SP in O and S'P in 0'. Then CO, which bisects both SP and SS', is parallel to PS'. Similarly CO' is parallel to SP. Hence each of the diameters CO, CO'.bisects chords parallel to tlie other, and the supplemental chords SP, S'P are parallel to these diameters. * Some of the signs may have to be changed according to the quadrants in which Q, q lie. b' t Or in the usual analytical form, tan 0. tan 0b - - + This proves again the result of Art. 70. T. 3 34 CHORD-PROPERTIES OF THE ELLIPSE. DEF. The circle described on the major axis as diameter is called the Auxiliary Circle. 72. PIOP. V. If the ordinate NP of any point P on tle ell'se beproduced to meet the auxiliary circle in p, then PN: pN = CB: CA. For PN': AN. NA' = CB2: CA2, [54. and AN. NA' = pN2. [Euc. II. 35. Therefore PN2: pN2 = CB2: CA2. 73. Conversely, if the ordinate NP be produced to p in the ratio CA: CB, the locus of p will be the auxiliary circle. And if the ordinate pN of any point on a circle be cut in a constant ratio, the locus of the point of section will be an ellipse whose axes are in that ratio. 74. PROP. VI. To draw conjzuqate diameters. Take conjugate radii Cp, Cd of the auxiliary circle. These include a right angle. [Euc. III. 3. Therefore z pCN= complement of dCR = CdR. Hence the right-angled triangles pCN, CdR are similar. And, since also Cp= Cd, they are equivalent, so that pN= CR, and dR = CN. Let pN, dR lut the ellipse in P, D. Then PN. DR: pN. dR = CB2: CA2. [72. Hence PN. DR: CR. CN= CB2: CA2, from above. Therefore CP, CD are conjugate. [70. CHORD-PROPERTIES OF THE ELLIPSE. 35 75. Conversely, if CP, CD be conjugate, and the ordinates NP, RD meet the auxiliary circle in p, d, the angle pCd will be a right angle, and the triangles pCN, CdR will be equivalent, so that CR =pN, and CN= dR. Hence PN: CR = CB: CA = DR: CN. [72. 76. PROP. VII. The sum of the squares of conjugate diameters is constant. Let CP, CD be conjugate radii. Let the ordinates NP, RD meet the auxiliary circle in p, d. Then the triangles pCN, CdR are equivalent, so that CR=pN, and CN=dR. [75. Therefore CN2+ CR2 = CN2 + pN2=,CA, [Euc. I. 47. since the radii CA, Cd are equal. Also PN: CR = CB: CA = DR: C. [75. Hence PN2 + DR2: CB2 = CN2 + CR2: CA2. But CN2+ CR2 = CA2, from above. Therefore PN2 + DR2 = CB2. By addition, CN2 + PNY2 + CRB + DR2 = CA2 + CB2; or CPT + CD2 = CA2 + CB2. 77. PROP. VIII. The parallelogram whtclhi as its sides equal and parallel to conjugate diameters is of constant area. For PN:: pN= CB: CA =DR: dR, as in Art. 74. Silence the rectilinear areas PCN, DCR, PDRN, pCN, dCR, pdRN, are each to eacli in the ratio of fB to CA. But A PCD = PDRN- PCN- D CR, and p Cd = pdRN - p CN- dCR. Hence A P CD is to p Cd, that is 2 CA, as CB to CA. 36 CHORD-PROPERTIES OF THE ELLIPSE. Therefore P CD = CA. CB. Describe a parallelogram by drawing parallels to conjugate diameters POP', DCD' through their extremities. Its area is four times that of the parallelogram PD; and the parallelogram PD = 2 A PCD = CA. CB. P 78. PROP. IX. If CP, CD be conjugate radii, SP. S'P = CD2. p -/ C Since (SP + S'P)2 = (2 C4)2, [57. therefore 2SP. S'P+ SP2 + S'P2 = 4 CA2. [Euc. II. 4. And because C bisects SS', therefore SP2 + S'P2 = 2C'S2+ 2CUP2. [Introd. Hence SP. S'P = 2 CA- CS2- CP' = CA4 + CB' - CP2 [60. = CD2. [76. CHORD-PROPERTIES OF THE ELLIPSE. 37 ~79. PROP. X. The focal ordinate is a harmonic mean between the segments of any focal chord. Let a focal chord PQ meet the directrix in R. Then SP: PR=SQ: QR, [45. or PR: QR = PR- SR: SR- QR. That is to say, PR, SR, QR are in harmonical progression, the first being to the third as the difference between the first and second to the difference between the second and third. But by parallels, if L denote the semi-latus rectum, PR: SR: QR=PM: SX: QN =SP: L: SQ. [Def. Therefore SP, L, SQ are in harmonical progression. ~80. The result may be written in either of the forms 1 1 2 SP 'SQ L' SP: SQ=SP-L: L-SQ, 2SP. SQ = L (SP+ SQ) = L. PQ. ~81. COR. If PQ, pq be any two focal chords, SP.SQ: Sp.Sq=PQ:pq. EXAMPLE. 93. Given the focus of a conic and a focal chord, find the lonus of the ends of the latus rectum. S8 CHORD-PROPERTIES OF THE ELLIPSE. ON AUXILIARY POINTS. DEF. If any principal ordinate NP be produced to p in the ratio CA CB, then p may be called the Auxiliary Point of P. When P lies on the ellipse, p lies on the auxiliary circle. [73. 82. If any number of points 0, P, Q... lie on a straight line which meets the axis in T, their auxiliary points o, p, q... will lie on a straight Une which meets the axis in the same point T. P Draw the ordinates pPN, qQM. Then from the definition of auxiliary points, and by parallels, pN: qM= PN: QM=NT: MT. Therefore Tpq is a straight line. And similarly it may be shewn that if O be any point on PQ, then o lies on pq. 83. If PQ be parallel to CD, then shall pq be parallel to Cd, where d is the auxiliary point of f. Draw the ordinate dDR. Then pN: d = PN: DR = NT: CR. Therefore pTN, d CR are similar triangles, so that Tp, Cd are parallel. CHORD-1ROPERTIES OF THE ELLIPSE. 39 84. PiloP. XI. If a chord PQ passes through a fixed point O, the rectangle OP.OQ varies as the square on the parallel radius CD. Take auxiliary points o, p, q, d, and let the straight lines opq, OPQ meet the axis iln. [82. Now in tlie triangle oTO, because pP is parallel to the side o 0, therefore OP: op= TP: Tp. Similarly, OQ: oq= TP: Tp. Compounding, OP.OQ: op.oq = TP: Tp2 = CD': Cd' by similar triangles TpP, CdD, their sides being parallel each to each. [83. Now since O is fixed its auxiliary point o is fixed. Therefore op. oq is constant, whether o be without or within the auxiliary circle. [Eue. mI. 35, 36. Hence, the radius Cp being constant, the ratio of OP. OQ to CD2 is constant whether O be without or within the ellipse. 85. A focal chordpSq varies as Sp. Sq or CD2. [81. 86. If PO Q, P'O Q' be any two chords* parallel to the radii CD, CD' and to the focal chords pq, p'q', then OP. O OQ: O. OQ'= C): CD'2 = pq: p''. [85. ~[87. COR. Hence, and from Art. 56, the chords of intersection of a conic with a circle are equally inclined to the axis. For if P, P', Q, Q' be concyclic these rectangles are equal. Hence pq, p'q' are equal, and therefore equally inclined to the axis. * As a particular case either of these chords may coincide with the parallel diameter. 40 CHORD-PROPERTIES OF THE ELLIPSE. 1~88. The following enunciation suits all conics: The rectangles contained by the segments of any two intersecting chords are to one another as the lengths of the parallel focal chords. We shall now give a more general definition of the terms Ordinate and Abscissa. DEF. Any two straight lines being taken as Coordinate Axes, from any point Q draw QV parallel to one of them and terminated by the other. Then QV is called the Ordinate and CVthe Abscissa of Q; and QV, CV together are called the Coordinates of Q. The axes here spoken of may or may not coincide with the axes of the ellipse. 89. PROP. XII. If QV, CV be the coordinates of any/ point Q on the ellipse referred to conjugate diameters DD', PP', then QV2: PV. VP' = CD: CP2. Complete the chord Q VQ' as in the figure on p. 44. Then QV. VQ': PV. VP'= CD2: CP, as in Art. 86. But CP bisects chords parallel to CD. Therefore QV2: PV. VP' = CD: CP2. We may also write QV2: CP2- CV2= CD: CP2. 90. If the axes of the ellipse be taken as coordinate axes, then (i) If the abscissoe be measured on the major axis we come back to Prop. Ir. (ii) If the abscissoe be measured on the minor axis, then QV2: PV. VP= CA: CBa 91. Hence it may be shewn by the method of Art. 72 that any ordinate Q V perpendicular to the minor axis is cut in the ratio CB: CA by the circle on that axis. The circle on BB' is in fact a Minor Auxiliary Circle, having auxiliary properties corresponding to those of the circle on AA'. CHORD-PROPERTIES OF THE ELLIPSE. 41 92. PROP. XIII. Tofind the length of any focal chord. Let PQ be a focal chord parallel to the radius CD. Denote by 2L the latus rectum, which is the focal chord parallel to CB. Then PQ::2L= CD: CB2 [85. = CD': L... [67. Hence PQ. CA= 2CD2. ~93. PROP. XIV. The straight ines which join adjacent extremities of any two focal chords meet two and two on the directrix. This will be proved in the course of Art. 105. T 94. The locus of the middle points of any system of parallel chords of a conic is a straight ine. And the bisecting Une meets the directrix on the straight lne through the focus perpendicular to the chords. This has been proved for the parabola in Art. 6; and the first part has been proved for the ellipse in Art. 68. We shall now indicate a method of proving the proposition generally. In the figure on p. 25 let PQ be one of a system of parallel chords. Draw a perpendicular S Y to PQ. Bisect PQ in O. Then since SQ: QR = SP: PR, [44. therefore SP2 - SQ2: PR' - QR2 = SP2: PR2. But SP2 and SQ2 are equal respectively to PY2 + SY2 and Q Y2 + SY2. By subtraction SP' - SQ2 = PY2- Q Y2 = PQ. 2 0 Y. [Introd. Also, PR2- QR2 = PQ. 2 OR. Hence O Y: OR = SP2: PR2, from above. Therefore O Y: OR is constant for parallel chords, [44. and the locus of O is a straight line meeting the directrix in the same point with SY. 42 CHORD-PROPERTIES OF THE ELLIPSE. 95. Let two diameters meet the directrix inp, d. Then if Cd bisects chords parallel to Cp, it may be easily shewn that S is the Orthocentre of tlhe triangle pCd, and hence that Cp bisects chords parallel to Cd. Also that pX. dX: CX2= CB: CA2, [64, 5. which agrees with Art. 70. ~96. To stew that a conic is concave to its axis. In the figure on p. 25, since SR bisects the angle supplenentary to PSQ, therefore if O be any point in the chord PQ, SO: OR < SP: PR. [Introd. But if O' be the point, above the axis, in which the ordinate of O cuts the conic, theil SO': OR = SP: PR. [44. Hence SO', being greater than SO, lies above it; or every arc PQ, however small, is more remote from the axis than the chord of the arc. ( 43 ) CHAPTER IV. TANGENT-PROPERTIES OF THE ELLIPSE. 97. THE tangents at the ends of any diameter are parallel to the ordinates of that diameter and to one another. Conversely, parallel tangents touch the curve at opposite ends of a diameter. [17. 98. The tangents at the ends of either axis are parallel to the other. 99. The sides of a parallelogram described as in Art. 77 are tangents. Hence The area of a circumscribing parallelogram which has its sides parallel to conjugate diameters is constant, and equal to the rectangle contained by the axes. 100. Let the normal at P meet DD' in F. Then PF. CD = parallelogram PD = CA. CB. 101. In Art. 86 it is shewn that the rectangles contained by the segments of any two intersecting chords are to one another as the rectangles contained by the segments of any other two chords parallel to the former. Hence if two of the chords, moving parallel to themselves, become tangents TP, Tp, and the other two chords become diameters DD' dd' parallel to those tangents, then the proposition assumes the form TP: Tp2= D2: Cd2. Hence any two tangents TP, Tp are as the parallel radii CD, Cd. 44 TANGENT-PROPERTIES OF THE ELLIPSE. 102. In Art. 82 suppose the points Q, q to become coincident with P, p. Then the lines PQ, pq become tangents to the ellipse at P and to the auxiliary circle at p respectively, and they always meet in a point T on the axis. Hence, the radius Cp being perpendicular to the tangent pT, CN. CT= C/p= CA. We proceed to prove a more general theorem which includes this. 103. PROP. XV. IfCV be the abscissa of a point Q, the tangent at which meets the radius of abscissoe CP in T, then CV. CT = CP2. The tangent at P is parallel to QV. Let it meet QT in R. Complete the parallelogram QRPO, viz. by drawing PO parallel to RQ. Then the diagonal RO bisects PQ. But PQ is a chord of contact, viz. of tangents RP, RQ; and therefore its bisector RO is a diameter. [22. Let it be produced to the centre C. Then by parallels CV: CP = CO: CR = CP: CT. Therefore CV. CT= CP2. 104. If P coincide with A, and N be written for F, then CN. CT= CA2. The corresponding property of the minor axis may be expressed, Cn. Ct = CB. TANGENT-PROPERTIES OF THE ELLIPSE. 45 ~105. PROP. XVI. Tangents at the ends of a focal chord meet on the directrix; and every tangent, measured from the curve to the directrix, subtends a right angle at thefocus. (i) Let Pp, Qq be any two focal chords; and let PQ meet the directrix in R. Then from the definition, SP: SQ=PR: QR; [45. or R lies on the bisector of z pSQ. [Euc. VI. A. So too it may be shewn that pq meets the directrix on the bisector of z pSQ. That is to say, PQ, pq meet the directrix in the same point R. (ii) Now let Pp, Qq be adjacent chords. And let Qq turn about S until it coincides with Pp. Thus the joining lines PQ, pq, which always meet on the directrix, become the tangents at P, p. And since SR is always equally inclined to Sp, SQ, therefore in the limit, when SQ coincides with SP, it makes equal angles with Sp, SP. That is to say, SR is at right angles to Pp, the chord of contact of tangents from R. Compare the next figure. ~106. Conversely, the chord of contact of the tangents drawn from any point on the directrix passes through the focus. ~107. DEF. The point of intersection of the tangents at the ends of any chord is said to be the Pole of the chord; and the chord of contact of the tangents from any point is said to be the Polar of the point. 46 TANGENT-PROPERTIES OF THE ELLIPSE. 108. PROP. XVII. If from any point T on the tangent at P to a conic, perpendiculars TL, TN be letfall on SP and the directrix, then SL: TN=SA: AX. As in Art. 28, SL: SP=R T: RP = TN: PM. Alternately, SL: TN= SP: PM = SA: AX. ~109. COR. Conversely, to draw tangents to a conic from a given point T. With. radius SL determined from the proposition describe a circle about S. Draw tangents TL, TJM to the circle. Then SL, SM meet the conic in the required points of contact. 110. PROP. XVIII. The focal radius to the pole T qf any chord of a conic makes equal angles with the radéi to its ends P, Q* With the construction of Art. 29, SL: T2N= SA: AX, since T lies on the tangent at P. And SM: TN= SA: AX, since T lies on the tangent at Q. Ience, in the right-angled triangles SLT, SMT, the sides SL, SM are equal. And ST is common. Therefore the angles TSL, TSM are equal, or ST makes equal angles witl SP, SQ. ~111. DEF. The point in which a chord meets the directrix is called the Foot of the chordt. ~112. The focal radii to the foot R and thepole T ofany chord PQ include a right angle. For SR, ST bisect supplementary angles pSQ, PSQ. [105, 110. * The enunciation of Art. 29 applies in all cases except when the tangents are drawn to opposite branches of a hyperbola. It will appear in the sequel that tangents so drawn subtend supplementary angles at either focus. t This definition is borrowed from Mr H. G. Day's treatise on the Ellipse. TANGENT-PROPERTIES OF THE ELLIPSE. 47 113. PROP. XIX. In the ellipse the tangent at P is equally inclined to SP, S'P; and the normal at P bisects the angle SPS'. (i) Let a perpendicular through P meet the directrices in M, 1'; and let R, R' be the feet of the tangent. [111. Then SP: S'P= PM: P ' [Def. PR: PR'; by similar triangles PMR, PM'R'. Also z PSR = a right angle = PS'R'. [105. Therefore in the triangles SPR, S'PR', the angles at P' are equal. That is to say, the tangent at P makes equal angles with SfP, S'P; or, as in fig. Art. 115, bisects the angle between SP produced and S'P. (ii) In the next figure let tPT be the tangent and PG the normal. Then z tPG = a right angle = TPG, or z SPt + SPG = S'PG + S'PT. Therefore, subtracting the equal angles SPt, S'PT, SPG=S'PG, or the normal bisects the angle SPS'. 48 TANGENT-PROPERTIES OF THE ELLIPSE. 114. PROP. XX. The circle through the.foci and any point P on the ellipse meets the minor axis on the tangent and the normal at P. t Let the circle through SPS' cut the minor axis in g, t. Then the equal arcs gS, gS' subtend equal angles at P. Therefore Pg is the normal at P. [113. Also gt, which bisects SS' at right angles, is a diameter of the circle, and therefore subtends a right angle at P. Hence Pt, being at right angles to the normal, is the tangent at P. 115. PROP. XXI. If T be thepole of the chord PP', then z STP = S'TP'. /Y jiT Let SP', S'P intersect in O. Produce SP to Q. Then TP, TS bisect the angles S'PQ, PSP'. [113, 110. TANGENT-PROPERTIES OF THE ELLIPSE. 49 1 i Hence z TPQ - TSQ = 2OPQ- P; 2 2 or z STP= POS. [Euc. I. 32. Similarly z S'TP' = OS'. 2 Hence, the vertical angles at O being equal, z STP= S'TP' ~116. PROP. XXII. If the normal to a conic at P meet the axis in G, then SG: SP=SA: AX..t - S t Let the tangent at P meet the directrix in R. Then the circle on PR as diameter passes through M the foot of the perpendicular from P to the directrix. It also passes through S, since PSR is a right angle. [105. Now PG is at right angles to PR and touches the circle. [Euc. II. 16. Cor. Therefore z SPG = SMP in the alternate segment. And z PSG = SPM, by parallels. Hence the triangles SPG, SMP are similar, so that SG: SP= SP: PM =SA: AX. T. 4 150 TANGENT-PROPERTIES OF THE ELLIPSE. 1117. Conversely, when P is given this relation determines G, and the normal at P may be drawn. 118. In the ellipse, SG SP= SP: PM= CS: CA. Also SG: PM= CS2: CA'= CS: CX. [64. Conversely, if SG, drawn as in the figure, satisfies this relation, then PG is the normal at P. Hence the following construction for the normal at P: Le cur PM on t Let fali a perpendicular PM on the directrix, and let MS meet the minor axis in g. Then Pg is the normal. For let it eut the axis in some point G; and let the minor axis cut PM in n. Then the parallels SCG, MnP are cut in the same ratio, so that SG: PM= CS: n~M = S: CX. 119. Also, by parallels and from what precedes, Pg: Gg = PM: SG = CX: CS. Therefore Pg: PG= CX: SX = CA2: CB. [64,. TANGENT-PROPERTIES OF THE ELLIPSE. 51 120. Again, by similar triangles PGNY, gPn, and since Pn= CN, NU G: CN= P: Pg= CB2: CA2, or the subnormal varies as the abscissa. ~121. PROP. XXIII. If from the points G, g in which the normal at P meets the axes, perpendiculars GK, gk be let fall on SP, then will PK* and Pk be constant. (i) Draw the principal ordinate PN. Then by similar right-angled triangles SKG, SN-P, SK: SN = SG:SP = SP: NX. [116. Hence SP-SIT: NX-SN= SP: NX, or PK is to SX as the eccentricity. Therefore PK = latus rectum; [Def. or PK.CA= CB2. [67. (ii) Also PK: Pk =PG: Pg, by parallels, therefore PK. CA: Pk. CA = CB: CA'. [119. And the antecedents being equal, the consequents are equal. Therefore Fk CA. EXAMPLES. 94. Shew that NK is parallel to SM, and deduce that PKI: SX=SP: PM. 95. If gk, gk' be perpendiculars to SP, SP', then Pk, Sk are equal respectively to Pk', S'k'. Deduce that Pk (SP + PS') CA. 96. If a circle has double contact with a conic, a chord of the circle thruugh the focus and either point of contact has one of two constant values. 97. If it passes through one focus determine the point of contact; and if through both foci, the eccentricity. * The first part only of this proposition applies to all oonics. The second part applies to central conics. 4_. 52 TANGENT-PROPERTIES OF THE ELLIPSE. 122. PROP. XXIV. The perpendicularsfrom the two foci of an ellipse upon any tangent contain a constant rectangle. Let fall perpendiculars SY, S'Y' and SZ, S'Z' upon two tangents which meet in a point T. Then z STY= S'TZ'. [115. Hence by similar triangles STY, S'TZ', and by similar triangles STZ, S'TY', SY: S'Z'= ST: S'T =SZ: S'Y'. Let TZ become parallel to the axis; then SZ, SZ' become each equal to CB. Silence always SY: CB = CB: S'Y', or SY.S'Y'= CB2. 123. PtoP. XXV. The feet of the focal perpendiculars on any tangent to an ellipse lie on the auxiliary circle. Let SY, S'Y' be perpendiculars on the tangent at P. TANGENT-PROPERTIES OF THE ELLIPSE. 53 Let SP, S'Y' meet in S. Then the tangent PY' bisects the angle S'Ps. [113. Hence the triangles sPY', S'PY' have their angles at P equal, and also their angles at Y. And the side PY' is common. Therefore sP = S'P, and s Y'=S' Y'. Hence CY' bisects both sS' and SS'. Therefore CY' is parallel to Ss and equal to Ss. Now Ss = SP Ps = SP + PS', from above, =2 CA. [57. Therefore CY' = CA; or Y' lies on the auxiliary circle. So too does Y. 124. Conversely, the straight line drawn from any point Y on the auxiliary circle at right angles to SY touches the ellipse. 125. Hence every focal chord meets the auxiliary circle in points Y, Z which lie on parallel tangents; and the rectangle contained by the focal perpendiculars SY, SZ on parallel tangents is equal to AS. SA' or CB2. [61. 126. Hence another proof of Prop. 24, since evidently SZ= S'Y'. 127. COR. Since CY', SP are parallel, therefore, if the diameter parallel to PY meet SP in k, Pk = CY'= CA. 128. Hence k is the foot of the perpendicular let fall on SP from the point in which the normal at P meets the minor axis. [121. 129. PROP. XXVI. Tangents at right angles meet on a fixed circle. Draw focal perpendiculars SY, S'Y' and SZ, S'Z'-on two tangents which intersect at right angles, viz. in T. The feet of tliese perpendiculars lie on the auxiliary circle. [123. 1Now CT2 = CA2 + TZ. TZ, as in Eue. III. 36. 54 TANGENT-PROPERTIES OF THE ELLIPSE. And the opposite sides of the rectangles ST, S'T being equal, TZ. TZ'= SY. S' Y' = CB2. [122. Therefore CT2 = CA2 + CB2, or the locus of T is a circle. 130. DEF. The circle which is the locus of the intersection of tangents at right angles is called the Director Circle. 131. PROP. XXVII. If the normal at P meet the axes in G, g, and CD be the radius conjugate to CP, then PG: CD = CB: CA, and Pg: CD=CA: CB. Use the figure of Art. 72, and supply the line PGg. Draw Pn perpendicular to the minor axis. (i) Then, since CD is parallel to the tangent at P, it is perpendicular to the normal. Therefore z PGN= complement of DCR = CDR. Hence, by similar right-angled triangles PGN, CDR, PG: CD=PN: CR = CB: CA. [75. (ii) Again, the triangle gPn is similar to PGN or CDR. Hence Pg: CD=Pn: DR = CN: DR, by parallels, =CA: CB. [75. 132. Hence PG. Pg = CD2 TANGENT-PROPERTIES OF THE ELLIPSE. 55 133. Also, as has been proved, PG: Pg= CB2: CA2. [119. 134. PROP. XXVIII. If the normal at P cut the axes il in G, g, and the diameter conjzuate to CP in F, then PF. Pg = CA2, and F.PG = CB2. k\ / (i) Let the diameter conjugate to CP meet SP in k. Then gk is perpendicular to SP. [128. Also, since the diameter CF is parallel to the tangent at P, it meets the normal at right angles, viz. in F. Hence PF. Pg = Pk2 = CA2. [12. (ii) And, since the angles at K, F are right angles, tlih points G, K, k, F are concyclic. Therefore PF. PG = PK. P = CB2. [121. 135. Or thus: Let the tangent at P meet the axes in T, t. Let fall perpendiculars PN, Pn on the axes, and produce them to meet the diameter CF in M, m. Then the angles at N,, n being right angles, F, G, N, M are concyclic; and F, g, m, n are concyclic. Hence PF. PG = PN. PLf= Cn. Ct, by parallels, = CB2. [104. And PF. Pg = Pn. Pn = CN. CT, by parallels, = CA.2 [104. 56 THE ELLIPSE. EXAMPLES. 98. The sides AD, DC of a rectangle ABCD are divided into the same number of equal parts, and straight lines are drawn from B, A respectively to the points of section. Shew that corresponding lines in the two series meet on an ellipse whose axes are equal to the sides of the rectangle. 99. Shew how to draw equal conjugate diameters. [74. 100. Apply Art. 94 to prove that a central conic is symmetrical with respect to its minor axis. 101. If SBS' be a right angle what is the eccentricity? 102. When is the angle SrS' a maximum 103. The segments of a focal chord subtend equal angles at X. 104. If chords PR, QR be produced to meet the directrix in p, q, the angle between the focal radii to p, q will be equal to half the angle between the focal radii to P, Q. 105. Two ellipses whose major axes are equal have a common focus; prove that they intersect in two points only. 106. The major axis is the maximum chord of an ellipse, and the minor axis its least diameter. 107. Given one focus of an ellipse, a point on the curve, and the length of the axis; shew that the loci of the other focus and of the centre are circles. 108. The circle inscribed in the triangle, SPS' touches SP in Jf, and SS' in N. Prove that P'M= A'S, and AM = SP. 109. What is the locus ofthe centre of a circle which touches two fixed circles. 110. Two circles have their centres fixed and the sum of their radii constant. Find the locus of the centre of a circle of given radius which touches them both. 111. The common chord of two ellipses having the same focus passes through the intersection of the corresponding directrices. 112. Given an ellipse, shew how to find C and S. EXAMPLES. 57 113. Given S' and a chord through S, find S. 114. Two conjugate diameters being given in position and magnitude, determine the axes. 115. The locus of a point which cuts parallel chords of a circle in a given ratio is an ellipse having double contact with the circle. 116. The circle on a focal radius touches the auxiliary circle. 117. The common tangents of the auxiliary circle and the circles on SP, S'P intersect on the ordinate of P. 118. Conjugate diameters meet the directrix at distances from the axis which contain a constant rectangle. 119. A focal cord and its diameter meet the directrix at distances from the axis which contain a constant rectangle. 120. The centre of a focal chord traces a similar ellipse. 121. The centre of a chord which cuts the axis in a fixed point describes an ellipse. [53. 122. A straight line equal to the radius of a circle slides with one end on a fixed diameter and the other end P on the convex side of the circumference. Shew that the coordinates of a point Q in the line vary as those of P, and hence that Q traces an ellipse. 123. Shew that SP - CA varies as the abscissa of P, and that (SP - CA)2 + (CA - SD)2 = CS2 [76. 124. The parallelogram described on conjugate diameters as diagonals is of constant area. 125. A diameter of an ellipse varies inversely as the perpendicular focal chord of the auxiliary circle. 126. If the radii CP, CQ be at right angles, 1 1 1 1 CP CA+ Q = - + C 127. The sum of conjugate diameters is a maximum when they are equal. 128. When is the sum of conjugate diameters a minimum? 5~a~8 ~ THE ELLIPSE. 129. SQ, S'Q being perpendiculars to conjugate diameters, the locus of Q is a concentric ellipse. 130. The perpendiculars to the axes from the points in which a common diameter meets the two auxiliary circles intersect two and two on the ellipse. [91. 131. The common diameters of equivalent and concentric ellipses are at right angles. L87. 132. If PP', DD' be conjugate diameters, then PD, PD' are proportional to the diameters parallel to them. 133. The sum of conjugate focal chords is constant. 134. A point in a straight line which slides between two fixed straight lines at right angles traces an ellipse. 135. Parallel diameters ofsimilar and similarly situated conics bisect the same systems of parallel chords. 136. 5With the pole of a chord as centre a circle can be described touching the four focal radii to the ends of the chord. 137. A circle can be drawn through the foci and the intersections of auy tangent with the tangents at the vertices. 138. The equi-conjugate diameters coincide in direction with the diagonals of a rectangle formed by the tangents at the ends of the axes. 139. The central perpendicular on the tangent varies inversely as the conjugate diameter. 140. If the tangent and ordinate at Q meet PP' the diameter of abscissoe in T, V, shew by Art. 101 that TP: TP'=PV: VP'. 141. Shew directly by the method of Art. 103 that TC. T = TP. T.. 142. If a chord of a conic subtends a constant angle at the focus, its envelope and the locus of its pole are conics having the same focus and directrix. [108. 143. The vertex of a circumscribed triangle whose base subtends a constant angle at the focus is a couic. EXAMPLES. 59 144. Shew by equating the angles of the quadrilaterals SPTP', S'PTP' to eight right angles that the external angle between the tangents is equal to half the sum of the angles which their chord of contact subtends at the foci. [115. 145. What is the corresponding theorem when the direction of PP' falls between the foci. 146. Any tangent meets parallel tangents on conjugate diameters. 147. If the tangent at P meets parallel tangents in Q, R, the rectangle PQ. PR is equal to CLD. 148. The polar of a point on the directrix passes through S. 149. The radii from a focus to the ends of a diameter make equal angles with the tangents at those points. 150. The straight lines joining the feet and the poles of any two chords subtend equal angles at the focus. [111. 151. The intercept on a tangent by parallel tangents subtends a right angle at the focus. 152. If s, s' be the reflexions of S, S' with respect to the tangent at P the triangles SPs', sPS' uill be equivalent. 153. A parallel to SP from S' meets SY on a circle. 154. Shew that SY. CD= SP. CB. 155. Also that SY2: CB2=SP: 2CA-SP. 156. The ordinate bisects the angle YNY', and the points Y, N, C, Y' are concyclic. 157. Shew that PG is bisected by SY', and by S'Y; and that it is a harmonic mean to SY, S'Y'. 158. Tangents being drawn from any point on a circle through the foci, shew that the bisectors of the angles between them pass through fixed points. 159. If the tangent and normal meet either axis in T, G, then CG. CT =CS. 160. The bisectors of the angles between the tangents from any point are tangent and normal to the confocals through that point. 60 THE ELLIPSE. 161. In Art. 114 shew that PG. Pg is equal to SP. PS', and deduce the theorems of Art. 131. [78, 119. 162. If CP be conjugate to the normal at Q, CQ will be conjugate to the normal at P. 163. If the tangent at P and its diameter meet the axis and directrix in T, D, then DTl is parallel to PS. 164. Shew how to draw tangents to an ellipse from a point on the auxiliary circle, or from any other external point. 165. The pole ofthe tangent at P with respect to the auxiliary circle lies on the ordinate of P. 166. If TP, TP' be the tangents in the first case, SP will be parallel to P'T. 167. A circumscribing parallelogram which bas two corners on the directrices has the other two on the auxiliary circle. 168. If an ellipse inscribed in a triangle has one focus at the orthocentre, the other focus will be at the centre of the circumscribed circle. 169. If an ellipse slides between two straight lines at right angles the locus of its centre is a circle. 170. The straight line joining the foci subtends at the pole of a chord half the sum or difference of the angles which it subtends at the extremities of the chord. 171. The portion of a normal chord intercepted between the directrices subtends at the pole of the chord half the sum of the angles which the straight line joining the foci subtends at the extremities of the chord. [112. 172. If a chord be produced to meet the directrices, the parts produced will subtend equal angles at the pole. 173. Supplemental chords equally inclined to the curve have their poles on the director circle. What is the corresponding property of the parabola' 174. The sum of two chords thus drawn is constant. 175, The normal at A is equal to L. [121. ( 61 ) CHAPTER V. CHORD-PROPERTIES OF THE RECTANGULAR HYPERBOLA. 136. THE hyperbola has been defined as a conic whose eccentricity exceeds unity. When the eccentricity is equal to J/ * the hyperbola is called Rectangular, and also Equilateral, for reasons which will appear. [155. 137. In the next figure, let S be the focus, and X the point in which the axis meets the directrix. Take C in SX produced, such that CX= SX. On the axis measure CA, CA' mean proportionals to CS, CX. It will be seen from Prop. I. that A, A' are the Vertices, or points in which the curve cuts the axis. The point C, which bisects AA', is the centre. 138. DEF. The circle on AA' is called the Auxiliary Circle. Its diameter BB' perpendicular to AA' is called the Conjugate Axis, and AA' is called the Transverse Axis. These are also, though equal, called the Minor and Major Axis. 139. Since CS= 2 CX = 2 SX, by construction, and CA2= CS. CX, therefore CS2 = 2 CA2, and CA2 = 2CX = 2SX2. 140. The square of the semi-latus-rectum is from the definition equal to 2SX', and therefore to CA'. Hence the latus-rectum is equal to the axis. 141. Also SA.SA'= CS-CA2= CA. [139. * The ratio of the a-agonal to the side Qta square. 62 CHORD-PROPERTIES OF 142. PROP. I. If PN be the principal ordinate of any point P on the curve, then CN2 PN2 = CA2. p/ pz Since PN2 + SVN2 = SP2 = 2NX2, [Def. and CN2 + SNV2 = 2 CX2 + 2NX2, [Euc. II. 10. Xbeing the middle point of CS, therefore CN2 - _PN2 = 2 CX2 = CA2. [139. 143. If Cn be the abscissa of P measured along the minor axis, then Pn2 - Cn2= CA2. 144. Since PN2 = CN2 - CA2 the ordinate PN is equal to the tangent NT fiom N to the auxiliary circle. Hence the circle described about N as centre, with radius NT, cuts a perpendicular to the axis through N in points P, P' which lie on the curve. Thus, by taking successive positions of N, we may determine any number of pairs of points on the curve. THE RECTANGULAR HYPERBOLA. 63 145. On the.form of the curve. It is evident from the proposition that the least value of CN is CA. Hence no part of the curve can lie between the tangents at A, A' to the auxiliary circle. Through C draw straight lines CE, CE', each inclined at half a right angle to the axis. Let EN, the ordinate of any point E on one of these lines, cut the curve in P. Then, since EN= CN, and from the proposition PV is less than CN, therefore PN is less than EN. Hence the curve lies wholly within the angle ECE' and the angle vertically opposite. And it spreads out to an infinite distance from both axes at once, since as PN increases from zero to infinity CN increases from CA to infinity. Like the ellipse, it is symmetrical with respect to both axes, since wlien the lengths PN, CNT at any point P of the curve are given, we may measure NP either upwards or downwards from the axis, and CN either to the right or to the left from C. Hence the curve has a second focus S' the Reflexion of S, and a second directrix the Reflexion of the first. [55. 146. It is evident that diameters and focal chords equally inclined to the axis are equal. 147*. PROP. II. The deference of the focal radii to any point on the curve is constant. Witli the same construction as above, S'P- SP: NX' - NX= SA: AX, or S'P -SP: 2CX =SA: AX. Therefore the difference of S'P, SP is constant. 148. Let P coincide with A. Then the constant value of S'P SP assumes the form S'A - SA, that is AA' or 2 CA. 149. Hence SP2 + S'P2 - 2S'P. S'P = 4CA2 = 2 CS. [139. Therefore 2 CS + 2SP. S'P= S'P2 + SP2 = 2 CP + 2 CS2, [Introd. since C bisects SS'. Therefore SP. S'P = CP2. * This applies as it stands to the hyperbola generally. 64 CHORD-PROPERTI ES OF 150. DEF. A line to which a curve approximates continuously and indefinitely, but whicli it does not meet at a finite distance, is called an Asymptote of the curve. 151. PROP. III. Tofind the asymptotes of the rectangular hyperbola. Through C draw two straight lines each inclined at half a right angle to the axis; and let them meet the principal ordinate PNof any point P on the curve in E, E. Suppose E to lie on the same side of the axis with P. [Fig. Art. 142. Then CN = EN= E'N. Hence EP. E'P= EN2PNV2 = CNV2-PN = CA2. [142. Therefore as E'P increases EP decreases; and it is evident that it may decrease continuously and indefinitely with the increase ot CN. Hence CE is an asymptote. So too is CE'. 152. In like manner it may be shewn that if an ordinate Pn of tlie minor axis meet the asymptotes in e, e', then eP. e'P= CA2. [143. 153. COR. The intercepts EP, E'P, and likewise eP, e'P, are equal. 154. COR. From P, which may be any point on the curve, draw perpendiculars Px, Py to CE, CE'. Then in the isosceles right-angled triangles PxE, PyE', 1 1 Px=2 =PE2, and Py2= '2. 2 2 Hence Px. Py = PE. PE' =2 CA. 155. The asymptotes are at right angles. Hence the name Bectangular Hyperbola. The Ilame Equilateral Hyperbola has reference to the equality of the axes. [138. THE RECTANGULAR HYPERBOLA. 65 156. PROP. IV. If a chord QQ' always parallel to itself cuts the asymptotes in R, R', the rectangle QR. QR' is constant. Draw QEE', parallel or perpendicular to the axis according as Q, Q' do or do not lie on opposite branches, and let it meet the asymptotes in E, E'. Then the triangle EQR is always similar to itself. So too is E'QR'. Therefore QR varies as QE, and QR' varies as QE'. Hence QR. QR' varies as QE. QE', or CA', and is therefore constant. [152. 157. COR. Hence QR. QR'= QR. Q'R'. Therefore the intercepts QR, Q'R' are equal, QQ' being any chord. 158. COR. The diameter bisecting QQ' bisects RR'. EXAMPLES. 176. If AA' be a diameter of a circle and PQ one of its ordinates, then will AP, A'Q intersect on a rectangular hyperbola. 177. Tangents to a parabola which include half a right angle intersect on a rectangular hyperbola. 178. If a parallel to an asymptote meet the curve in P and a principal double ordinate Qq in 0, then QO.Oq= 4 OCP. T. 5 66 CHORD-PROPERTIES OF 159. PROP. V. The locus of the middle points of any system ofparallel chords is a straight line through the centre. Q Let Qq be any one ofa system of parallel chords; CM Let Qq be any one of a system of parallel chords*; CM, Cm the abscissoe of its extremities. Let O be the centre of the chord, and OL its ordinate. Draw qK parallel to the axis to meet QM. Then QM12 + CA2 = CM2, [142. and qm2 + CA2 = Cm. Therefore QM2 - qm2 = CM 2 - m2, or QM-qm: CM+ Cmn = CMI- Cm: QM +qm. Therefore 2 OL 2 CL = qK: QK, [Introd. But the ratio qK: QK is constant for parallel chords. Therefore also OL: CL is constant, and the locus of O is a straight line through C. 160. PROP. VI. Conjugate diameters make complementary angles with either axis. For OL: CL = qK: QK, as in Prop. v., = nM: QM, if n be the point in which Qq meets the axis. * Whether meeting one branch only or both branches. t See the first note on p. 33. THE RECTANGULAR HYPERBOLA. 67 Hence the right-angled triangles QnM, COL are similar, so that z QnM= COL = complement of O CL; or if CD be parallel to qQ, then CO, CD make complementary angles with CA, and therefore also with CB. [Fig. p. 68. 161. COR. Conjugate radii CP, CD are equally inclined to the axes each to each. 162. COR. Also they are equally inclined to either asymptote. 163. Hence it appears that the acute angle between two conjugate diameters lies wholly in one of the four quadrants bounded by the axes. Also that one of every two conjugate diameters makes with the axis an acute angle greater than half a right angle, and therefore evidently does not meet the curve. Likewise one of every two diameters at right angles does not meet the curve. 164. Every diameter which makes with the axis an acute angle less than half a right angle meets the curve. For, in the figure on p. 62 draw a diameter cutting the intercept EP in O. Then however small an angle the diameter makes with CE, the length EO may be increased indefinitely by making the ordinate recede from C. Therefore EO becomes at some point equal to and then greater than EP, which diminishes continually. That is to say, the diameter cuts the curve. 165. Hence it appears that no straight line through C except CE and CE' can be an asymptote. In like manner it may be shewn that no other straight line can be an asymptote. Also that a straight line parallel to an asymptote cuts the curve once and does not meet it again. 5 2 68 CHORD-PROPERTIES OF 166. DEF. It is convenient to define the length of a diameter which does not meet the curve as equal to the length of its conjugate*; but we shall avoid speaking of the "extremities" of a diameter which does not meet the curve. 167. PROP. VII. If CP, CD be conjugate radii, and PN, DR the ordinates of P, D, the triangles PCN, DCR will be equivalent. For z PCN = complement of DCR [160. = CDR. Hence the triangles are similar. And their sides CP, CD are equal. Therefore they are equivalent. 168. COR. Since CN= DR, and CR = PN, therefore CN2 - CR2 = DR2 - PN2 = CN2 - PN2 = CA2. [142. 169. COR. Also DR2 - CR2 = CN2 -PN2 = CA2, where CD may be any radius which does not meet the curve. * In the next figure a circle is drawn to indicate this equality. This must not be confounded with the auxiliary circle. THE RECTANGULAR HYPERBOLA. 69 170. PROP. VIII. The parallelogram * which has its sides equal and parallel to conjugate diameters is of constant area. In the last figure let CP, CD intersect in O. Then since D CR = PCNX, [167. subtracting COR, A OCD = PORN, and adding POD, A PCD = PDRN, or 2 APCD = (DR + PN) NRt = (CN+ CR) (CN- CR) = CA2. [168. For the rest compare Art. 77, and see fig. Art. 200. 171. PROP. IX. The angle between two diameters ws equal to that between their conjugates. For if CE be an asymptote, and CP, CD; Cp, Cd conjugate radii, such that PCE = DCE, [162. and p CE=dCE, by subtraction z PCp = D Cd. 172. More generally: The acute or obtuse angle between any two chords is equal to the acute or obtuse angle between any two chords conjugate to the former each to each. [70. 173. Since by Art. 71 supplemental chords as PQ, P'Q and PQ', P'Q' are conjugate, therefore the angles between PQ, PQ' are equal to those between P'Q, P'Q'. Hence any chord QQ' subtends at the ends of any diameter PP' angles which are either equal or supplementary. It may be shewn that the angles subtended are equal or supplementary according as the direction of QQ' falls within or without PP'. The next figure shews one of the four cases which occur. * In this case a rhombus. f For PDRN is made up of the triangles DRN, PND. 70. CHORD-PROPERTIES OF 174. PROP. X. If CV be the abscissa measured on the diameter PCP' of a point Q on the curve, then QV = PV.P'V CV2- CP2. a Produce Q V to meet the curve in Q'. Then Q Q' subtends at P, P' angles which, not being equal, are supplementary. [173. On CV produced take Vp equal to VP. Then since Q Q, Pp bisect one another they are the diagonals of a parallelogram. Therefore z QpQ' = QPQ = supplement of QP'Q'. Hence p, Q, P', Q' are concyclic, [Euc. III. 22. and QV2 =pV. P'. [Euc. III. 35. Therefore Q V2=PV. P'V= CV2-CP2. 175. Hence another proof of Prop. IV.: On Q Q' take VR, VR' each equal to CV. Then QR. QR' =RV2- QV= CV2- QVa = CP2 whence it may be shewn that CR, CR' are the asymptotes. 176. If Cv be the abscissa of Q measured parallel to QQ', then evidently Qv is equal to CV, and Cv to QV. Therefore Qv2 - CP2= Cv2. THE RECTANGULAR HYPERBOLA. 7I 177. PROP. XI. Diameters at right angles are equal. In the figure on p. 68 draw CP' at right angles to CD below the axis, and let CP be conjugate to CD. Then z D CA + PCA = a right angle [160. = P' CA + DCA, by construction. Ience the angles PCA, P'CA are equal. Therefore CP' = CP= CD. [146. 178. The ordinates in Prop. vII. have been taken to be principal ordinates. More generally, suppose CA to be any radius: then since the lines CP, CD and CIV, DR are conjugate, therefore z PGN= CDR. [172. Hence tle triangles PCN, CDR, having also the angles at C, N equal and the sides CP, CD equal, are equivalent. The corollaries then follow with the help of Prop. x. EXAMPLES. 179. In the rectangular hyperbola diameters at right angles bisect chords at right angles. -180. The bisectors of the angles between supplemental chords of a rectangular hyperbola are parallel to fixed lines. 181. Given a diameter and one other point of a rectangular hyperbola, construct the curve. 182. If on an arc AB of a circle whose centre is O there be taken points P, Q such that are AP= 2 arc BQ, then a rectangular hyperbola described on AO as diameter so as to pass through the intersection of OB with the tangent to the circle at A, will also pass through the intersection of AP, OQ. 183. Shew how to trisect a given circular arc by describing rectangular hyperbolas. 72 CHORD-PROPERTIES OF 179. PROP. XII. If a chord QQ' passes through a Jixed point 0, the rectangle QO. OQ' varies as the square of the parallel radius CP. Let V be the centre of the chord; Cq the radius towards 0; Cv the abscissa of q. Then Q V2- CP2 = CV, [176. and qv- CP2=Cv. Therefore QV'-CP2 2 q - P2 = CV: Cv2 = 0 V qv', by parallels. Hence OV2- QV'+ CP2: CP2= 0V2: qv But the right-hand ratio is equal to CO' Cq2, which is constant since 0, q are fixed points. Therefore the left-hand ratio is constant. Hence OV- QV': CP is a constant ratio. Therefore QO. OQ' varies as CP. 180. The proof is similar when Q, Q' lie on the same branch. When CO does not meet the curve the above result may be obtained with the help of Art. 178. THE RECTANGULAR HYPERBOLA. 73 181. Hence the rectangles contained by the segments of any two intersecting chords are as the squares of the parallel radii. 182. PROP. XIII. If PSQ be the focal chord parallel to the radius CD, then SP. SQ = CD2. For, the axis itself being a focal chord, SP. SQ: CD2= SA. SA': CA, [181. which is a ratio of equality. [141. 183. COR. Focal chords at right angles are equal. [177. 184. If QQ', RR' be chords at right angles, meeting in O, then QO. OQ'=RO. OR'. [177. The points Q, Q', R, R' are so situated that they cannot lie on one circle. Hence no exception to Art. 87 arises. 185. Since QO: OR = OR': OQ', therefore QR' is perpendicular to Q'R, and QR to Q'R'. Hence, if any triangle RQ'R' be inscribed in a rectangular hyperbola, its Orthocentre Q will lie on the curve. 186. Conversely, a conic which passes through the orthocentre and the vertices of a triangle must be a rectangular hyperbola; for it has three pairs of equal diameters at right angles, and it is evident that it cannot be a circle. 187. It may be shewn that the centre of a rectangular hyperbola described about a triangle lies on the nine-point circle, since the diameters of the hyperbola drawn to the middle points of the sides contain two and two the same angles as the sides. [172. 188. To prove again that Q in Art. 185 is on the curve: The straight line joining the middle point of RR' to V, the middle point of QQ', being a diameter of the nine-point circle, subtends a right angle at G. [187. Ience CVbisects chords at right angles to RR'. Therefore Q is on the curve. ( 74 ) CHAPTER VI. TANGENT-PROPERTIES OF THE RECTANGULAR HYPERBOLA. 189. IN the figure on p. 62 supposepP' to be any chord, and P such a point that the tangent thereat meets the chord at right angles in a point T. Then, since by a limiting case of Art. 184, TP2= Tp. TP', therefore Pp, PP' are at right angles. HIence the following construction for the tangent at P: Draw any two chords Pp, PP' at riqht angles. Then the perpendicularfrom P on pP' is the tangent required. 190. The rectangular hyperbola has no tangents at right angles. For if the tangents from a point T be produced beyond the curve to meet the asymptotes in O, 0', then 00', subtending a right angle at C, cannot subtend a right angle at T. [Euc. I. 21. 191. The portion Tt of any tangent intercepted by the asymptotes is bisected at the point of contact P. For by Art. 157, if a chord Qq meets the asymptotes in R, r the intercepts QR, qr are equal; and when the chord becomes the tangent at P these intercepts become PT, Pt. [Fig. Art. 200. Hence PT = Pt = CP = CD. [166. 192. Hence the tangents at the ends of a diameter P, P' meet the asymptotes at the vertices of a rhombus TTt't, which is of constant area 4CA2. [170. 193. Also QR. QR' becomes equal to PT2 or CD)2 TANGENT-PROPERTIES OF RECTANGULAR HYPERBOLA. 75 194. Art. 172 includes the case of tangents regarded as limiting cases of bisected chords. Thus, in the next figure, QT being the tangent at Q, since CQ, QT are conjugate, andalso CV, QV, therefore z QCV= TQ V= QtC. 195. PROP. XIV. If CV, QV be the coordinates of a point Q on the curve, and if the tangent at Q meet the radius ofabscissoe CP in T, then CV. CT = CP2. Q For since z QCV= TQV, [194. the triangles Q CV, TQV, equiangular at V, are similar, so that CV: QV=QV: VT, or C V. VT= Q V2. Hence CV. CT= CV2-QV2 = CP2. [174. 196. When the radius of abscissse does not meet the curve the points V, T assume positions v, t on opposite sides of C. 'The same method applies. 197. If the tangent at Q meets any two conjugate diameters in T, t, the rectangle QT. Qt is equal to CQ2. For the triangles QCT, QtC have their angles at Q, Q; C, t; T, C equal. [194. 76 TANGENT-PROPERTIES OF 198. PROP. XV. If the normal at P meets the axes in G, g, then PG= CP = Pg. The normal cuts the diameter conjugate to CP at right angles. Hence, F being the point of section, z PGC= complement of FCG = PCG. [160. c (f Therefore PG = CP = Pg, similarly. 199. COR. Draw perpendiculars GK, gk to SP. Then Pk = PK= CA. [121, 140. 200. PROP. XVI. Any tangent contains with the asymptotes a triangle of constant area. THE RECTANGULAR HYPERBOLA. 77 For if the tangent at P meet the asymptotes in T, t, and CD be conjugate to CP, then since the triangles CPT, CPD are between the same parallels, and since P bisects Tt, [191. TCt 2CPT= 2 CPD = CA2. [170. EXAMPLES. 184. The subnormal is equal to the principal abscissa. 185. The circle on SS' meets the asymptotes on the tangents at A, A'. 186. The circle through C which' has its centre at P on the curve passes through the points in which the tangent and normal at P meet the asymptotes and the axes respectively. 187. Draw a tangent to a rectangular hyperbola which shall be parallel to a given line. 188. What is the locus of the middle point of a line which cuts off a constant area from the corner of a square? 189. If two concentric rectangular hyperbolas be such that the axes of one are the asymptotes of the other, they will intersect at right angles. 190. If two concentric rectangular hyperbolas touch the same straight line, the lines joining their points of intersection to their respective points of contact subtend equal angles at the centre. 191. If a right-angled triangle be inscribed in a rectangular hyperbola, the hypotenuse will be parallel to the normal at the opposite angle. 192. If two rectangular hyperbolas touch one another their common chords through the point of contact will include a right angle, and the remaining common chord will be parallel to their common tangent. 193. A circle through the centre and two points of a rectangular hyperbola passes also through the intersection of the lines drawn from each of the two points parallel to the polar of the other. ( 78 ) CHAPTER VII. CHORD-PROPELTIES OF THE HYPERBOLA. IN dealing with the Hyperbola generally we shall have not so much to investigate a new set of propositions as to observe the particulars in which the curve under consideration differs from the ellipse, or, as regards asymptote-properties, from the rectangular hyperbola. 201. Let S be the focus of the hyperbola, and X the point in which the axis meets the directrix. The vertices A, A' satisfy the relation SA: AX= SA' A'X, each of these ratios being equal to the eccentricity. Since the eccentricity is greater than unity it is evident that A, A' lie on opposite sides of X. From the above proportion we deduce that SA + SA' AX + A'X = SA - SA X A'X, each of these ratios being also equal to SA AX. Hence, if C be the middle point of AA', 2CS: 2CA=2CA: 2CX - SA:AX. 202. With the same construction as for the ellipse it may be shewn that in fig. 2 Z ZSZ'= a right angle, [49. and that PN2: AN. NA'= SX2: AX. A'X, [53. which is a constant ratio. CHORD-PROPERTIES OF THE HYPERBOLA. 79 203. The square of the ordinate varies as NA. NA', or as the square of the tangent from N to the Auxiliary Circle. [144. 204. If we define CB as that value of PN which results from making AN and A'N in the above proportion each equal to CA, then we may write, analogously to Art. 54, PN2: ANV. NA' = CB: CA, or PN2: CN- CA2= CB2: CA; but in this case B is not apoint on the curve. 205. The method of Art. 145 may now be applied mutatis mutandis to determine the form of the curve; the main difference being that the lines CE, CE' are to be taken subject to the relation EN2: CN2= CB2: CA2; whence it appears that, as before, PN is less than EN, and the curve lies wholly within the angle ECE' and that vertically opposite. 206. DEF. The hyperbola inay be called Acute or Obtuse according as the angle ECE' is acute or obtuse; or, in other words, according as CB is less or greater than CA. DEF. On a perpendicular through C to AA' take equal lengths CB, CB. The lines AA, BB' are called the Transverse Axis and the Conjugate Axis respectively. The former is also called the Major Axis, although it may be less than BB', and the latter is called the fMinor Axis. 207. Through A draw a perpendicular to the axis, and let it meet the lines CE, CE' in H, h. Then from the construction in Art. 205, AH= CB=Ah, [Fig. 1. and C02= CA2 + CB. 80 CHORD-PROPERTIES OF THE HYPERBOLA. 208. As in the first and second parts respectively of Art. 67, it may be shewn that if 2L be the latus rectum of the hyperbola L. CA = CB2 and L. CA = CS. SX. Hence CB2= CS. SX, and CA2 + CB = CS. CX+ CS. SX [201. = CS2. Also SA. SA'= CS2-CA2 = CB2. 209. Using the method of Art. 68 and the figure on p. 66, we may deduce from Art. 204 that OL. DR: CL. CR= CB2: CA2. Hence, if CP, CD be conjugate radii, and CN, CR the abscisso of P, D, then PN. DR: CN. CR= CB2: CA. [70. Or we may apply Arts. 94, 5. 210. DEF. The rectangular hyperbola described on AA' as axis may be called the Auxiliary Hyperbola. A diameter of this curve may be called, with reference to an acute or obtuse hyperbola on the same axis, an Auxiliary Diameter. 211. If the ordinate NP of any point on the hyperbola meet the auxiliary hyperbola in p, then, as in Art. 72, PN: pN= CB: CA. 212. To draw conjugate diameters. Take auxiliary conjugate radii Cp, Cd, and let the ordinates pN, dB meet the hyperbola in P, D. [Fig. 3. Then PN. DR: pN. dR= CB: CA2. [211. Hence, the triangles p CN, d CR being equivalent, PN. DR: CR. CN= CB2: CA2. [167, 8. Therefore CP, CD are conjugate. [209. CHORD-PROPERTIES OF THE HYPERBOLA. 81 213. DEF. If Cd does not meet the auxiliary curve, CD also will not meet its curve. But we may define CD as terminated by the ordinate of d. 214. Conversely, if CP, CD Le conjugate, PN: CR = B: CA = DR: CN. [75. 215. Hence DR' - PN: CB2 = CNV - CR2: CA' But CN- CR2- CA2; [168. therefore DR - PN2 = GB2. Hence CN+~ PiN'- CR' - DB2 = CA2- CB2, or CP2 - CD2 = CA'- CB; that is to say, the difference of the squares of conjugate diameters of a hyperbola is constant. 216. As in Art. 77, the rectilinear areas [Fig. 3. PCN, DCR, PDRN, p CN, dCR, pdRN, are each to each in the ratio of CB to CA. But a PCD = PD-RN+D CR -P CN, and A pCd = pdRN + dCR - pCN; therefore A PCD is to p Cd, that is 1 CA2, [170. as CB to CA. Hence the triangle PCD is equal to ~ CA. CB, and the parallelogram which has its sides equal and parallel to conjugate diameters is equal to AA'. BB'. [Fig. 1. 217. Art. 84 applies to the hyperbola, if instead of Euc. III. 35, 6 Art. 179 be assumed. 218. Also QV2: PV. VP' = CD: CP2. [89. Hence QV2: CV2 - CP = CD: CP2, antd Q V+ CD2: CV2 = CD2: CP2. 219. Analogously to Art. 204, CD2 is the value of QV2 which would result from equating PV and P'V to CP. T. 6 82 CHORD-PROPERTIES OF THE HYPERBOLA. THE ASYMPTOTES. 220. Through P (fig. Art. 200) draw a parallel to QV, and take PT, Pt each equal to CD. Let CT, Ct cut QV in R, r. Then BV2: CV= PT': Cp2 = CD': CP2. Hence R V2 = Q V+ CD. [218. Therefore RQ. Qr = R V - QV2 = CD2. And since RQ diminishes indefinitely as the ordinate recedes from the axis, therefore CR, or Cr, is an asymptote. [165. 221. From this construction it appears that if a straight line meets the curve in Q, q and the asymptotes in R, r then since the same diameter bisects Qq and Br, the intercepts QR, qr are equal. Also it appears that as Qq moves parallel to itself, the rectangle BQ. Qr has the constant value CD2. This constant value becomes CB2 when Qq is parallel to the conjugate axis, and CA2 when it is parallel to the transverse axis. 222. If from any point Q on the curve, Qx, Qy be drawn each parallel to one asymptote and terminated by the other, then Rr being supposed to move parallel to itselt; one of the lines Qx, Qy, say Qx, varies as QR, and the other Qy varies as Qr. But RQ. Qr is constant. Therefore Qx. Qy is constant. [220. When Q is at A, Qx and Qy become ~ CEh and J CH. [207. Hence always Qx. Qy = (CA2 + CB") = CS. [208. ( 83 ) CHAPTER VIII. TANGENT-PROPERTIES OF THE HYPERBOLA. 223. FROM the construction of Art. 220, it follows that the portion Tt of any tangent between the asymptotes is bisected at the point of contact. [Fig. Art. 200. 224. The triangle TCt is of constant area, being equal to 2CPD, or CA. CB. [200, 216. 225. The triangles TCt, HCh being equal, their sides about the angle C are proportional, so that CT: CH= Ch: Ct. [Fig. 1. Therefore CT. Ct = CI. Ch = CS2. [207, 8. 226. Art. 103 applies to the hyperbola when CP meets the curve*. If V, T assume positions v, t on the conjugate diameter DD', which does not meet the curve, then Ct: QV= CT: VT. [Fig. p. 75. Hence, Cv being equal to QV, Cv. Ct: QV= CV. CT: CV. VT = CP: CV2- CP, by the first case. Therefore Cv. Ct = CD2. [218. Fig 5 suits the axis-property in an acute hyperbola. Proof: CN: C = CO: CR = CA: CT. Therefore CN. CT= CA2. * Arts. 195, 6 may be extended by Projection. 84 TANGENT-PROPERTIES OF THE HYPERBOLA. 227. In Art. 110, it is true in all cases that TSL = TSM. When the tangents are drawn as in fig. 4, to opposite branches of a hyperbola, the one angle is supplementary to that which TP subtends at S, and the other is equal to that which TQ subtends at the same point. Hence TP, TQ subtend supplementary angles at S. In like manner it may be shewn that the angle TS'P is equal to the supplement of the angle TS' Q. 228. The tangent at P bisects the angle SPS'. [Fig. 4. The proof in Art. 113 is applicable. Or we may proceed as follows in the case of either curve: Since PR subtends riglit angles at M, S, the points P, 31, R, S are concyclic, and [105. z SPR - SMX. Similarly z S'PR'= S'M'X'. Hence, the triangles SMX, S'M'X' being equivalent, SPR = S'PR'. 229. The normal at P, being at right angles to PR, is equally inclined to the focal distances SP, S'P. 230. Art. 114 applies to the hyperbola, except that Pg is in this case the direction of the tangent, and Pt that of the normal. If a confocal ellipse and hyperbola intersect in P, heir tangents at P will be at right angles. 231. If TP, TP' touch opposite branches of a hyperbola it may be 3hewn that STP = S' TP. [115. If the tangents be drawn to the same branch these angles may ue shewn to be supplementary. TANGENT-PROPEITIES OF THE HYPERBOLA. 85 232. Arts. 122-8 apply mutatis mutandis to the hyperbola. Let SY, S'P meet in s. Then it may be shewn that 2 CY= S's = S'P- SP [Fig. 8. =2 CA. [147. Hence the locus of Y, or Y', is the auxiliary circle. [138. Let Z be the point diametrically opposite to Y' in which SY produced meets the circle. Then since evidently SZ= S' Y', SY. S'Y'= SY. SZ= SA. SA' = CB2. [208. 233. It may be shewn that in an acute hyperbola tangents at right angles intersect on a fixed circle. For if T be the point, between Y, Y', in which two sucli tangents intersect*, then A2 - CT = TY. TY' [Euc. nI. 35. = CB, as in Art. 129. 234. The method of Art. 131 applies to the hyperbola. [Fig. 7. 235. Art. 135 applies to the hyperbola. [Fig. 6. 236. So too does the preceding Article. 237. Various minor Articles in preceding chapters apply with or without modification, as the case may be, to the general hyperbola; and many Examples too which have been stated for the ellipse only, apply also to the hyperbola. The student should make a practice of considering in every case, including that of the parabola, whether a theorem is applicable mutatis mutandis to more than one of the three species of Conics. EXAMPLES. 194. The centre of an equilateral hyperbola described about an equilateral triangle is on the inscribed circle. * The curve in fig. 8 is not drawn to suit this case. 86 THE HYPERBOLA. 195. In the rectangular hyperbola SPS' is a right angle when CP2 = 2CA2. 196. A focal perpendicular on an asymptote of a rectangular hyperbola is equal to the semi-axis. What is the corresponding property of the general hyperbola 197. CY being drawn perpendicular to the tangent at P, the triangles PCA, CA Y are similar. 198. The intersections of any two conjugate chords of a rectangular hyperbola with the asymptotes cannot be concyclic. 199. A chord of a rectangular hyperbola meets the asymptotes in R, r and passes through a fixed point 0. Shew that RO. Or varies as the square of the parallel diameter. 200. A rectangular hyperbola confocal with an ellipse cuts it at the ends of the equal conjugate diameters. 201. If CP, CD be conjugate radii of a rectangular hyperbola, then will D be the reflexion of P with respect to one of the asymptotes. 202. Ellipses being inscribed in a parallelogram, their foci lie on a rectangular hyperbola. 203. If lines be. drawn from any point on a rectangular hyperbola to the ends of a diameter the difference of the angles which they make with the diameter will be equal to the angle which it makes with its conjugate. 204. From fixed points A, B straight lines are drawn intersecting in a point C such that the difference of the angles C-BA, CAB is constant. Find the locus of C. 205. A conic through the four common points of two rectangular hyperbolas is itself a rectangular hyperbola. 206. A conic through the centres of the four circles which touch the sides of a triangle is a rectangular hyperbola, and its centre is on the circumscribing circle. 207. Any chord of a rectangular hyperbola subtends equal or supplementary angles at the ends of a perpendicular chord. 208. On opposite sides of a chord of a rectangular hyperbola equal segments of circles are described. Shew that the four points in which the circles meet the curve again are the vertices of a parallelogram. EXAMPLES. 87 209. The tangents to a rectangular hyperbola at the vertices of an inscribed triangle meet two and two on the lines joining the feet of the perpendiculars of the triangle. 210. If each vertex of a triangle be the pole of the opposite side with respect to an equilateral hyperbola, the circumscribing circle will pass through the centre of the hyperbola. 211. Straight lines joining the ends of conjugate focal chords meet on the asymptotes. 212. A circle and a rectangular hyperbola intersect in four points. If one of their common chords is a dianeter of the hyperbola, the other is a diameter of the circle. 213. A parallelogram ABCD has its diagonal AC at right angles to the side AB. If CD be divided into any number of equal parts and straight lines be drawn from A to the points of section, and if AC be divided into the same number of equal parts and straight lines be drawn from B to the points of section, then will corresponding lines in the two series meet on a hyperbola. 214. Hence shew how to construct a rectangular hyperbola. 215. If N be a point in AA' produced the circles described about S, S' with radii AN, A'N meet on the hyperbola. What is the corresponding construction for the ellipse? 216. The difference of the focal distances of any point is greater or less than the axis according as the point lies on the concave or the convex side of the hyperbola. 217. If SY be perpendicular to the tangent at P SY2: CB =SP: 2CA +SP. 218. From a fixed point 0, OP is drawn to a given circle. Find the envelope of a straight line through P inclined at a constant angle to OP. 219. The tangent from N to the circle on XX' varies as the normal. 220. In a central conic a circle through P and either G or g cuts off from the focal distances lengths whose sum is constant. 221. Given in an ellipse a focus and two points, the other focns describes a hyperbola. 88 THE HYPERBOLA. 222. If P, Q be points on a central conic a confocal passes through the intersections of SP, S'Q and SQ, S'P. 223. The tangents at these points and at P, Q cointersect. 224. Given the asymptotes and a point on the curve, find the foci 225. In the hyperbola SP is equal to a line drawn from P parallel to an asymptote to meet the directrix. 226. If two hyperbolas have the same asymptotes a chord of one touching the other is bisected at the point of contact. 227. The straight lines joining the points in which two tangents meet the asymptotes are parallel. 228. The common chords of a circle and a hyperbola make equal angles with the asymptotes. 229. The second tangents to a central conic from the points in which a tangent meets conjugate diameters are parallel. 230. If the tangent at P meets conjugate diameters in T, t, the triangles SPT, S'TP will be similar. 231. A hyperbola can be drawn through the ends of any two radii of an ellipse so as to have the conjugate diameters as asymptotes. 232. If PP', DD' be conjugate diameters of a hyperbola and Q any point on the curve, shew that QP2 + QP'2 exceeds QD2 + QD" by a constant quantity. 233. Given two points of a parabola and the direction of its axis, the locus of the focus is a hyperbola. 234. The centre of the inscribed circle of the triangle SPS' lies on the tangent at the nearer vertex. 235. A chord which subtends a right angle at the vertex meets the axis in a fixed point. 236. Find the locus of a point which divides the part of any tangent between the asymptotes in a constant ratio. 237. If a hyperbola touches the sides of a quadrilateral inscribed in a circle and if one focus lies on the circle, the other lies on the circle. ILSJ})OHS UJUJE[I7?-Y 0F' (0N1 US. 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