A GEOMETRICAL TREATISE ON CONIC SECTIONS. i A GEOMETRICAL TREATISE CO IC S E CTIOSON CONIC SECTIONS. WITH NUMEROUS EXAMPLES.,or tie VThr of itl001oo anb if1nbunft il tje lnibersities. BY THE E V. HW. H. D EW, M. A. ST. JOHN'S COLLEGE, CAMBRIDGE, PROFESSOR OF MATHEMATICS IN KING'S COLLEGE, LONDON. FOURTH EDITION. llnborn aab (ambribc: MACMILLAN AND CO. MDCCCLXIX. [The Right of Transla.tion aind Rrprloduction is i','sw'''.] LONDON: R. CLAY, SONS, AND TAYLOR, PRINTERS, BREAD STREET HILL. CONIC SECTIONS. INTRODUCTION. 1. DEF. The curve traced out by a point, which moves in such a manner that its distance from a given fixed point continually bears the same ratio to its distance fiom a given fixed line, is called a Conic Section. The fixed point is called the Focus, and tile fixed line the Directrix. Thus if S be the focus, and KK' the directrix, and P a point from which PM is drawn at right angles to the directrix, the curve traced out by P will be a Conic Section, provided P M move in such manner that SP always bears the same ratio to PM. K S (1.) When the distance from the fixed point is equal to the distance from the fixed line, that is, when SP is equal to PJ, the Conic Section is called a Parabola. K' (2.) When the distance from the fixed point is less than the distance from the fixed line, that is, when the ratio wliich B 12 CONIC SECTIONS. SP bears to P1M is less than unity, the Conic Section is called an Ellipse. (3.) When the distance from the fixed point is greater than the distance from the fixed line; that is, when the ratio which SP bears to PMl is greater than unity, the Conic Section is called an Hyberbola. 2. The reason of the term Conic Sections being applied to these curves is that, when a Clone is intersected by a plane surface, the boundary of the section so formed will, in general, be one or other of these curves. I purpose to investigate the properties of the Conic Sectiotns from the definitions given above, and afterwards to show in what manner a Cone must be divided by a plane in order that the curve of intersection may be a Parabola, Ellipse, or Hyperbola. CHAPTER I. THE PARABOLA. PROP. I. 3. The focus and directrix of a parabola being given, to find any number of points on the curve. Let S be the focus, and TKK' the directrix. Draw XSx at right angles to the directrix, line SX in A; then and bisect the since A S = AX,.'. A is a point on the curve. The point A is called the Vertex, and the line Ax, with respect to which the curve is evidently symmetrical, is called the Axis. B2 4 CONIC SECTIONS. On the directrix take any point M; join SM; and draw MP at right angles to the directrix. At the focus S make the angle M1SP equal to the angle SMP; then SP= PM,. P is a point on the curve. So by taking any number of points, Mi', Ml", on the directrix, we may obtain as many points, P', P", on the curve as we please, and the line which passes through A and all these points will be the parabola whose focus is S and directrix KK'. CoR. 1. As M is taken further away from the point X, the line SM and the angles SMIP, iMSP, and, consequently, the lines SP and PM, continually increase. Hence, since XM and AlP increase together, the curve recedes at the same time both from the axis and directrix; and since the angle SMP can never exceed a right angle, and the lines SP and MP will therefore always meet, it is evident that there is no limit to the distance to which the curve may extend on both sides of the axis. COR. 2. The parabola may be described practically in the following manner. K M}'1- _ - __= Q A I x Let S be the focus and KX be the directrix; and let a rigid bar QM, having a string of the same length as itself fastened at one end Q, be made to slide parallel to the axis with the other end Mon the directrix; then if the other end of the string be fastened at the focus, and the string be kept stretched by means of the point of a pencil at P, in contact with the bar, since SP will always be equal to PM, it is evident that the point P will trace out the parabola. CONIC SECTIONS. 5 PROP. II. 4. The distance of any point inside the parabola from the focus is less than its distance from the directrix; and the distance of any point outside the parabola from the focus is greater than its distance from the directrix. AM Q. /P A. iA ' (1.) Let Q be a point inside the parobola. Draw QM1 at right angles to the directrix, parobola in P; join SP; then since SP = PM,. SP andPQ= Q. But SP and PQ> SQ,.. QM>SQ. (2.) Let Q be a point outside the parabola. meeting the Draw MQ at right angles to the directrix, and produce it to meet the parabola in P; join SP; then since SQ and QP > SP, and SP = PMll,. SQ and QP> P1M,.. S > Q1e. Col. Conversely a point will be inside or outside the parabola according as its distance from the focus is less or greater than its distance from the directrix. 5. DEF. The line PN (see fig. Prop. III.) drawn at right angles to the axis from the point P in the curve is called the Ordinate of the point P, and the line AN the Abscissa. The double ordinate BC drawn through the focus, and terminated both ways by the curve, is called the Latus Rectum. G~6 C)CONIC SECTIONS. PROP. III. Tile Latus Rectum B C = 4A S. p IV Draw BK at right angles to the directrix. Then SB =BK = SX = 2AS,..B C =AS. 6. DEF. If a point P' be taken on the parabola (see fig. Prop. IV.) near to P, and PP' be joined, the line PP' produced, in the limiting position which it assumes when P' is made to approach indefinitely near to P, is called the Tangent to the parabola at the point P. PrOP. IV. If the tangent to the parabola at any point P intersect the directrix in the point Z; then SZ will be at right angles to SP. Let P' be a point on the parabola near to P. CONIC SECTIONS. 7 Draw the chord PP', and produce it to meet the directrix in Z; join SZ. Draw PMl, P'M' at right angles to the directrix; join SP, SP'; and produce PS to meet the parabola in Q. Then, since the triangles ZMP, ZM'lP' are similar,.. ZP: ZP:: MP: 'P',:: SP: SP', SZ bisects the angle P'SQ. (Euclid, VI. Prop. A.) Now when P' is indefinitely near to P, and PP' becomes the tangent at the point P, the angle PSP' becomes indefinitely small, while the angle Q SP' approaches two right angles, and therefore the angle P'SZ, which is half of the angle P'S Q, becomes ultimately a right angle. Hence, when PZ is the tangent, the angle ZSP is a right angle, or SZis perpendicular to SP. Con. Conversely, if SZ be drawn at right angles to SP, meeting the directrix in Z, and PZ be joined, PZ will be a tangent at P. 8 8 ~~~~CONIC SECTIONS. Pn'op. V.T 7. The tangent at any point P of angle between the focal distance SP, P IV on the directrix. a parabola bisects the and the, perpendicular 1R Let the tangent at P meet the directrix in the point Z: join SZ; then since the angle Z SR is a right angle, (Prop...Zs 2~+SR2 =Pz'. AlIso ZMI2 + KRI = RZ1..ZS2 + SR2 =Z _M2 +MRJP Bnt SR PM1, Z S= ZM-1. Now in the triang-les ZPS, ZPiJJ7 ZR, PS = ZR, PRMI each to eac, and ZS = Z111, *the angle SRZ = the ang-le MRZ; or RZ bisects the angle SRJJL. CONIC SECTIONS. 9 CoK. 1. If ZP be produced to R, then the angle SP = the angle MPR. CoR. 2. It is evident that the tangent at the vertex A is perpendicular to the axis. PROP. VI. 8. The tangents at the extremities of a focal chord intersect at right angles in the directrix. Let PS Q be a focal chord, and let the tangent at P meet the directrix in Z. Join SZ; then the angle Z SP is a right angle, (Prop. IV.) and.. also the angle ZS Q is a right angle,. ZQ is the tangent at Q, (Prop. IV. Cor.) or the tangents at the extremities of the focal chord PSQ intersect in the directrix. Again, draw Pill, QM' at right angles to the directrix; then since MP, PZ = SP, PZ, each to each, and the angle MIPZ = the angle SPZ,. the angle MZP = the angle SZP,. the angle SZP is half of the angle SZM.7 So the angle SZQ is half of the angle SZM',. the angle PZQ is half of the two SZM1 and SZM. But the angles SZM and SZM' = two right angles,. the angle PZQ is a right angle, or the tangents at the extremities of a focal chord intersect at right angles in the directrix. 10 10 ~~~~CONIC SECTIONS. PROP. VII. 9. If the tangent at any point P of a parabola meet the axis produced in the point T1 and PNV be the ordinate of the point P, then NVT= 2ANIV i l l - I _ _ _ _ _ _ _ _ _ P '7 Join SP, and draw P111 at right angles to the directrix; then the angle SP T = the angle KRP T =the angle, S T1,.SI,= SP. But SP=PM31= XX;, STXX. But 'A S A X, the remainder A IT the remainder Az- X, NVT 2 ANIDEF. T le line NT is called the Subtain gent. 10. DEF. The line PC, drawn at right angles to PT, is called the Normnal at the point P, and NGC the Subnormal. PR'OP. VIII. If thie -normal at the point P of a parabola meet the axis in the point C, then NCG = 2-A S. CONIC SECTIONS. 1 I 1 -Since the angle SPG = the complement of the, angle SPT, and the angle SUP = the complement of the, angle STJP, and also the angle SPT =the angle STP, (Prop,. VIE) the ang-le SPU the angle SUP, SU= SP. lBut SP= PJVi- X.AT, Taking away the common part SIV, the remainder NU = SX =2A,_ S. PThop. IX. ii. If PNI be an ordinate to the, parabola at the point P; thenrPNV2= 4iAS. AX Since TP U is a right angle, and PAT perpendicular to TU; PAT is a meoan. proportional between TN and ATV U; or PN2 = TNAT. NU. (Euclidl, VI. 8 Gor.) But TNV = 2.A N, (Pr-op. VII.) and NU = 2A4 S, (Prop. VIIIL) PN' =4 A S. AN. PuZop. X. 12. If the tangent at any point P intersect the tangenit at the vertex in Y, thenr S Y will bisect PT at right angl1es, and will be a mean proportional between BA,4 anid SP. Draw PN at right angles to the axis; then since A Yis — parall el to P1V. TY: YJ):: TA: AN_/. But A T= AN4, (Prop). VII.). TY= PY; and -.-Y, YP= SY, YT, cachi to eachl, and SP = STI, (Prop). VII.) the angle S YP = the an gle SY T, B Y -is perpendicular to PT. 12 CONIC SECTIONS. Again, since TYS is a right angle, and YeA perpendicular to ST,. SY is a mean proportional between ST and SA; or SY2 = ST. SA. (Euclid, VI. 8 Cor.) But ST = SP, (Prop. VII.) S2-.'. S SA. CoR. If PM be drawvn at rtiht angles to the directrix, and lMY be joined, then since SP, P Y = MP, P Y, eac to each, and the angle SPY = the angle MP Y, (Prop. V.). the angle SYP the angle MYP,.. S Y and YM1 are in the same straight line. P'OP. XI. 13. To draw a pair of tangents to a parabola from an external point. K <_ CONIC SECTIONS. 13 Let 0 be the given external point. Join OS, and with centre 0 and radius OS describe a circle, cutting the directrix in il and iMl', which it will always do, on whichever side of the directrix 0 is situated, since 0 is nearer to the directrix than to the focus. (Prop. II.) Draw Mi Q and ll' Q' parallel to the axis meeting the parabola in Q and Q'. Join OQ, OQ'; these will be the tan ts required. Join S Q and S Q'; then OQ, QS = OQ, QM, each to each, and 08= OM,. the angle OQS = the angle OQ-M,.. Q is the tangent at Q. (Prop. V.) So 0 Q' is the tangent at Q'. PROP. XII. 14. If from a point 0 a pair of tangents OQ and OQ' be drawn to a parabola, the triangles OSQ, OSQ' will be similar, and OS will be a mean proportional between SQ and S Q'. Join SM, cutting OQ at right angles (Prop. X. Cor.) in the point Y; then since the angle SQO = the angle MQO, (Prop. V.) and the angle MQO = the angle SlMM', each of these angles being the complement of the angle QMY,.the angle SQ 0 = the angle SMMi'. But the angle SMM1' at the circumference is half the angle SOM' at the centre, and is therefore equal to the angle SOQ'.. the angle SQO = the angle SOQ'. So the angle SOQ = the angle SQ' O, the remaining angle OSQ = the remaining angle OSQ'. 14 CONIC SECTIONS. And therefore the triangle 0SQ is similar to the triangle 059'. SQ: SO::SO: SQ', SQ. AQ = 5Q 2. or SO is a mean proportional between SQ and 59. Pioop. XIII. 15. If a pair of tangents 09, 09' be drawn to a parabola, and 0 V be drawn parallel to the axis meeting Q Q' in V, then 9 Q' shall be bisected in V. I/ Vl~ Draw QM1, Q'M' at right angles to the directrix. Join OM, O1'; and let 0 V meet. MM' in Z Then, since 0O1 = 03', (Prop. XI.) the angle 0Zj = the angle 0M`' and the angle OZM = the angle OZ31', CONIC SECTIONS. 15 and the side OZ is common to the triangles OZM, OZJL1', iJiE =Z 1M'Z. And becanse the lines Q124 ZT7 Q'T' are parallel,. /QT VQT:: ilfz: 'Z But MZ - M31'Z, Q Q' is bisected in V. PROP. XIV. 16. If from a point 0 a pair of tangents OQ, OQ', be drawn to a parabola, and OV be drawn parallel to the axis meeting the parabola in P, and QQ9' in V, then the tangent at P will 'be parallel to Q Q' and 0 J7 will be bisected in P. P Draw the tangent R PR' meeting 0 Q, 0Q' in R and li'. Join P Q, and draw 1R W parallel to the axis, meeting P9 in W; 16 CONIC SECTIONS. Then, by the last Proposition, PW= WQ. And because R TV is parallel to OP,.O. o: RQ:: PW: WQ. But PWT= WQ,. OR = RQ; so OR' = R'Q,.. 0: R Q:: OR': R'Q',. R' is parallel to Q Q'. Again, since PR is parallel to Q V,. OP: PV:: OR: RQ. But OR= RQ,.OP =PV. COR. From this it is manifest that if any number of parallel chords be drawn in a parabola, their middle points will all lie on the line parallel to the axis whicl passes through the point where the tangent drawn parallel to the chords meets the parabola. DEF. Any line PV, drawn from a point P in the parabola parallel to the axis, is called a Diameter. The point P is called the Vertex of the diameter PV; and the tangent at P the Tangent at the Vertex. The diameter consequently bisects all chords parallel to the tangent at the vertex, and the tangrents at the extremities of any chord will intersect in the diameter corresponding to that chord. DEF. A line QV, drawn parallel to the tangent at P from a point Q in the curve, is called the Ordinate to the diameter PV. PROP. XV. 17. If QV be an ordinate to the diameter P V, then Q V =4. SP.PV. CONIC SECTIONS. 17 Q R Produce Q V to meet the parabola in Q'; and draw the tangents Q 0, Q'O, meeting VP produced in the point 0. (Prop. XIV.) Also let the tangent at P meet O Q in R, and join SP, SR, and SQ. Now since from the point B two tangents R P, B Q are drawn to the parabola, the triangle RPS is similar to the triangle S Q, (Prop. XII.).. the angle SRP = the angle SQR. But the angle SQR = the angle-ST Q, (Prop. VII.) = the angle POR,.'. the angle SRP = the angle POR, and the angle SPR = the angle OPR, (Prop. V. Cor. 1.).. the remaining angle R SP= the remaining angle ORP,.*. the triangle SPR is similar to the triangle POR,.. SP: PR:: P: PO,.P. 2= SP. PO, = SP. P V (Prop. XIV.) Again, since QV is parallel to PR,... QV: PR:: OF: OP. But 0 V= 2 OP, (Prop. XIV.) C 18 CONIC SECTIONS.. QV=2 PR, Q V = 4 PR2, =4 SP. PV. 18. DEF. The double ordinate to the diameter PV, drawn parallel to the tangent at P, and passing through the focus, is called the Parameter of the diameter P V. PROP. XVI. The parameter of the diameter PV = 4. SP. Draw QSQ' through the focus parallel to the tangent at P, and let the tangent at P meet the axis produced in T; then Q 2 = 4 SP. PV. (Prop. XV.) But PV = ST = SP, (Prop. VII.).Q. QV-2= 4SP2; or QV= 2SP,. '. QQ'= 4SP. CONIC SECTIONS. 19 PROP. XVII. 19. If two chords of a parabola intersect one another, the rectangles contained by their segments are in the ratio of the parameters of the diameters which bisect the chords. Let the chords Qq, Q'q' intersect one another in the point 0. Bisect Qq, Q'g' in Vand V'; and draw the diameters P 7, P' V' parallel to the axis. Also, through 0 draw OR parallel to PV; and through R draw R W parallel to Q V. Now, since Qq is divided equally in V and unequally in 0,.. QO. Oq = Q V2 - 0, (Euclid, II. 5) = QV2_ -RW, 4 SP. P - 4 SP. PW, (Prop. XV.) = 4SP. RO. So Q'O. Oq' = 4SP'. RO. Hence QO. Oq: Q'O. Oq':: 4SP: 4 SP'. c2 20 CONIC SECTIONS. By Euclid, IT. 6, the same may be proved to be true if the point O be without the parabola. PROP. XVIII. 20. If from an external point O a pair of tangents Q, O Q' be drawn to the parabola, and the chord Q Q' be joined, the area of the figure bounded by QQ and the curve is two-thirds of the triangle Q 0 Q'. r r Draw the diameter OV meeting the curve the tangent at P meet OQ, OQ' in R and E'. Join QP, Q'P; then in P; and let since OR = Q,.. the triangle OPR = - the triangle OPQ, = 1 the triangle VP Q. CONIC SECTIONS. 21 So the triangle OPR' = 2 the triangle VPQ',. the triangle ORR' = - the triangle PQQ'. Again, if through R and R' we draw the diameters Rp, RIp'; and at the points p and p' draw the tangents rpr,, rp'r', we can prove in the same manner as before that the triangle Rrr, = - the triangle QpP, and the triangle R'r'r', = the triangle Q'p'P. Continuing in this manner to form new triangles by drawing diameters at the points r, r,, and r', r,', and tangents at the points where these diameters meet the curve, we can prove that the exterior triangles formed by the tangents are the halves of the interior triangles formed by joining the points of contact with the extremities of the chords. And the same will hold however the number of the triangles be increased. Hence the sum of all the exterior triangles will be equal to half the sum of all the interior angles. Now when the number of the triangles is increased indefinitely, the sum of the exterior triangles will represent the exterior figure OQPQ', and the sum of the interior triangles the area of the interior figure QPQ. Ience the area of the figure O QP = Q - the area of the figure QP Q'.area of the figure OQPQ' = 1 the area of triangle QOQ',.area of the figure QP Q' = ~ the area of triangle Q Q'. 21. DEF. If with a point 0 on the normal at P as centre and OP as radius, a circle be described touching the parabola at P and cutting it in Q; then when the point Q is made to approach indefinitely near to P, the circle is called the Circle of Curvature at the point P. (Seefig. Prop. XIX.) 22 CONIC SECTIONS. PROP. XIX. The chord of the circle of curvature, at a point P of a parabola, drawn parallel to the axis = 4SP. Let PT be the tangent, and PG the normal at the point P. With centre 0 and radius OP describe a circle cutting the parabola in the point Q. Draw RQX parallel to the axis meeting the circle in X and the tangent at P in R. Also draw Q V parallel to PR, and P W parallel to the axis; then since RP touches the circle at P,. B Q. RX = PR2. (Eucid, III. 36.) CONIC SECTIONS. 2:3 But PR = QV2 = 4 SP. PV, (Prop. XV.).-. R. RX = 4SP. PV. But RQ = PV.. X. X = 4 SP. Now when the circle becomes the circle of curvature at F, the points B and Q move up to and coincide with P, and the lines RX and P W become equal. Hence the chord of the circle of curvature parallel to the axis = 4 SP. Cor. 1. If P U be the diameter of the circle of curvature, and PF the chord through the focus; then since the angle FPU = the angle VP U, (Prop. VIII.).P. PF = P I = 4SP. COR. 2. If SY be drawn at right angles to PT; then the triangle PFUis similar to SYP,.. PU: PF:: SP: SY, orPU: 4SP:: SP: SY. PEoP. XX. If QVQ' be any ordinate to the diameter PV, the circle described through the three points P, Q, Q' will intersect the parabola in a fourth point, which depends only upon the position of P. Draw the ordinate PN, and produce it to meet the parabola in P'; then, since the subtangent = 2. AN. (Prop. VII.) The tangents at P and P' will meet the axis in the same point T. Draw PR parallel to TP', meeting the parabola in R, and Q Q' in 0; then PO. OR: QO. OQ':: SP': SP. (Prop. XVI.) 24 CONIC SECTIONS. But SP = T = SP', (Prop. VII.). PO. OR= QO. OQ'. Hence by the converse of Euclid III. Prop. 22, the point R is on the circle which passes through P, Q, Q'. COR. I. Since TP and TP' are equally inclined to the axis, the lines Q Q', PR, which are parallel respectively to TP and TP', are also equally inclined to the axis. COR. 2. When the point V is brought indefinitely near to P, QQ' coincides with the tangent to the parabola at P, and becomes also a tangent to the circle at P, since Q and Q' are indefinitely near to each other. The circle therefore becomes the circle of curvature at the point P. Hence if PR be drawn parallel to the tangent at P, or be equally inclined to the axis with PT, it will meet the parabola in the point where the circle of curvature at P intersects the parabola. PROBLEMS ON THE PARABOLA. 1. The diameter of the circle described about the triangle BA C is equal to 5A S. (See fig. Prop. III.) 2. If from the point G, GK be drawn at right angles to SP, then PK= 2AS. (Seefig. Prop. VII.) 3. If the triangle SPG is equilateral, then SP is equal to the latus rectum. (See fig. Prop. VII.) 4. PQ is a common tangent to a parabola and the circle described on the latus rectum as diameter; prove that SP and S Q make equal angles with the latus rectum. 5. Prove that P Y. PZ = SP2, and that P Y. YZ = A S. SP. (See fig. Prop. VII.) 6. If PL be drawn at right angles to AP, meeting the axis in L, and PN be the ordinate of P, then NL = 4A S. 7. The tangent at any point P of a parabola meets the directrix and latus rectum produced in points equally distant fiom the focus. S. Prove that NAry= TY, and that TP. TY = TS. TV. (See fig. Prop. VII.) 9. If a circle be described about the triangle SPVT, the tangent to it from A = I PN (See fi. Prop. VII.) 10. If the ordinate of a point P bisect the subn6rmal of P', the ordinate of P is equal to the normal of P'. 11. If from any point on the tangent to a parabola a line be drawn touching the. parabola, the angle between this line and the line to the focus from the same point is constant. 26 CONIC SECTIONS. 12. A circle and parabola have the same vertex and axis. BA'C is the double ordinate of the parabola which touches the circle at A', the extremity of the diameter through the vertex A. PP' is any other ordinate of the parabola parallel to this, meeting the axis in V, and AB produced in R; prove that the rectangle P. RP' is proportional to the square of the tangent drawn from N to the circle. 13. Draw a parabola to touch a given circle at a given point, and such that its axis may touch the same circle in another given point. 14. If from the point of contact of a tangent to a parabola a chord be drawn, and another line be drawn parallel to the axis meeting the chord, tangent, and curve, this line will be divided by them in the same ratio as it divides the chord. 15. If the diameter PV meet the directrix in 0, and the chord drawn through the focus parallel to the tangent at P in V, prove that VP = PO. 16. Prove that the locus of the intersection of a diameter PV with the chord drawn through the focus parallel to the tangent at P is a parabola. 17. If a circle and parabola have a common tangent at P, and intersect in Q and R; and Q V, UR be drawn parallel to the axis of the parabola meeting the circle in V and U respectively, then VUis parallel to the tangent at P. 18. AB and A C are two lines at right angles to each other. From a fixed point C on A C, CR is drawn parallel to AB. On A R, produced if necessary, P is taken such that the perpendicular PN upon AB is equal to CR. Prove that the curve traced out by P is a parabola. 19. If from a point P of a circle PC be drawn to the centre; and R be the middle point of the chord PQ drawn parallel to a fixed diameter A CB, then the curve traced out by the intersection of CP and AR is a parabola. 20. If two equal tangents 0 Q, 0 Q', be cut by a third tangent, their alternate segments are equal. CONIC SECTIONS. 27 21. E is the centre of the circle described about the triangle O Q Q'. Prove that the circle described about the triangle QEQ' will pass through the focus. (See fig. Prop. XIII.) 22. PSp is any focal chord of a parabola. Prove that AP, Ap will meet the latus rectum in two points Q, q, whose distances from the focus are equal to the ordinates of p and P. 23. PSp is a focal chord of a parabola, RDr the directrix meeting the axis in D; and Q any point on the curve. Prove that if QP, Qp be produced to meet the directrix in R, r, half the latus rectum is a mean proportional between DR, Dr. 24. OP and 0 Q are two tangents to a parabola. On Q 0 produced, 0 Q' is taken equal to 0 Q; prove that OS. PQ'= OP. OQ. 25. If QD be drawn at right angles to the diameter PV, then QD= 4AS.PV. 26. If through any point 0 on the axis of a parabola a chord PO Q be drawn, and PM, QNT be the ordinates of the points Pand Q, prove that AM. AN= A 02. 27. If AP and A Q be drawn at right angles to each other from the vertex of a parabola, and PlI, QN be the ordinates of P and Q, prove that the latus rectum is a mean proportional between A M and A N. 28. OAP is the sector of a circle whose centre is 0. If the radius OA remain fixed while the angle A OP changes the centre of the circle inscribed in the sector, A OP will trace out a parabola. 29. QSQ' is a focal chord parallel to AP; PN, QM, Q'M' are the ordinates of P, Q, and Q'. Prove that SM'2 = AM. ANand that MM' = AP. 30. PQ, P Q' are drawn from any point P cutting the ordinates Q'V', Q V in R' and R, prove that VR is to V'B' in the triplicate ratio of Q V to Q' V'. 28 CONIC SECTIONS. 31. On a chord of a parabola as diameter a circle is described cutting the parabola again in two points. If these points be joined, the portion of the axis between the two chords is equal to the latus rectum. 32. If 0 Q, 0 Q' be a pair of tangents to a parabola, and the chord QQ' be a normal to the curve at Q, then 0 Q is bisected by the directrix. 33. Two equal parabolas having the same focus and their axes in contrary directions intersect at right angles. 34. The radius of curvature at the extremity of the latus rectum is equal to twice the normal. 35. If from any point P of a parabola PF and PH be drawn making equal angles with the normal PG, then S G = SF. SH. 36. If a triangle be inscribed in a parabola, the points when the sides produced meet the tangents at the opposite angles are in the same straight line. 37. If the tangents 0 Q, 0 Q' be cut by a third tangent in R, B', prove that OR: Q:: R'Q': OR'. 38. If from the vertex of a parabola chords be drawn at right angles to one another, and on them a rectangle be described, the curve traced out by the further angle is a parabola. 39. Prove that 2PY is a mean proportional between AP and the chord of the circle of curvature at the point P of the parabola drawn through the vertex A. (See fig. Prop. VII.) 40. If a circle described upon the chord of a parabola as diameter meet the directrix it also touches it, and all chords for which this is possible intersect in a point. 41. If a parabola roll upon another equal parabola, the vertices originally coinciding, the focus traces out the directrix,. 42. The circle of curvature at the extremity of the latus rectum intersects the parabola on the diameter of curvature passing through the point of contact. CHAPTER II. THE ELLIPSE. 22. DEF. The Ellipse is the curve traced out by a point which moves in such a manner that its distance from a given fixed point continually bears the same ratio, less than unity, to its distance from a given fixed line. (See Introduction.) PROP. I. The focus and directrix of an ellipse being given, to find any number of points on the curve. Let S be the focus, and MX the directrix. Draw SX at right angles to the directrix, and divide SX in the point A, so that SA may be to AX in the given fixed ratio less than unity; then A is a point on the curve. On XS produced take a point A', such that SA': A'X:: SA: AX; then A' will also be a point on the curve. On the directrix take any point M; and through 1M and S draw the line M YSY' meeting A Y and A' Y', drawn at right angles to AA' in the points Y and Y'. On YY' as diameter describe a circle, and draw MPP' parallel to A A', cutting the circle in the points P and P'; P and P' will be points on the ellipse. 30 3CONIC SECTIONS. Join PY, RY', BR; then since SY: YM:: BA: AX, (Euclid, VI. 2) and BY' Y'M:: BA': A'X, (Euclid, VI. 2) SY:YM S:::Y':Y'K; or, alternately, BY BY':: YM,: Y'I, and the angle YR Y' in a semicircle is a right angle, P Y bisects the angle SPM;* BR: PMl:: BY M, S.A:AX. So we may show that BR': P'Ml:: SY: YI,:: BA: AX. P and R' are points on the curve. * For, if not, make the angle YPs equal to YPM; then s Y YM:: sP: PM. (Euclid, VI. 3.) And since, if P Y bisect sPM, P Y', being at right angles to P Y, also bisects the angle sPM', s Y': YM:: sP: PM. (Euclid, VI. A.) Hence sY: YM::sY': Y'M, or sY: sY' YM: Y'M, the points S and s coincide. CONIC SECTIONS. 31 In the same way, by taking other points on the directrix, we may obtain as many more points on the curve as we please. COR. 1. Since, corresponding to every point P on the curve, there is a point P' situated in precisely the same manner with respect to A' Y' as P is with respect to A Y, it is clear that if we make A'S' equal to AS, and A'X' equal to AX, and draw X'M' at right angles to AX', the curve could be equally well described with S' as focus and M'X' as directrix. The ellipse is therefore symmetrical, not only with respect to the line AA', but also with respect to the line 0C drawn through the middle point of YY' at right angles to and bisecting AA'. CoR. 2. The line OP will bisect the angle SP S'. Let OP meet SS' in G. Produce MP to meet X'M' in M'. and draw Ol' passing through the focus S'; then SP: P:: S'P: PMP', or, alternately, SP: S'P:: PM: PM'. (1) Again, SG: PM:: S'G: PM', or, alternately, SG S:: P': PM', (2). from (1) and (2) SP: S'P:: SG: S'G,. PG bisects the angle SPS'. (Euclid, VI. 3.) It will be shown hereafter (Prop. XI.) that the normal to the ellipse at the point P also bisects the angle SPS'. Hence the ellipse and circle have the same tangent at the point P. The ellipse will consequently touch all the infinite series of circles which can be described in the same manner as the one in the figure by taking different points on the directrix. 32 CONIC SECTIONS. PROP. II. 23. If C be the middle point of AA'. then proportional between CS and CX, CA is a mean or CS. CX = CA. (See fig. Pro Since SA': A'X:: SA: AX. Alternately SA' SA:: A'X: AX. SA'+ SA:A::A'X+AS orAA': SA:: X': A. AA': XX'::SA: AX or CA: CX::A SA: AX. Again, SA' SA::A'X: A. A'-SA: SA::A'X-AA or SS' SA AA': AJ Alternately SS': AA':: SA: AX or CS: CA:: SA: AX Hence from (1) and (2) CA: CX:: CS: CA,. III.): AX; ~,. (1)* r: AX; (2).. CA2= CX. CS. or CA is a mean proportional between CS and CX. COR. Since the three lines CS, CA, CX are proportional, therefore, by the definition of duplicate ratio and Euclid, VI. 20 Cor. CS: CX: CS2: CA2. (3) PROP. III. 24. If P be any point on the ellipse, then SP + S'P = AA'. * N.B. The results'(1), (2), (3) should be remembered, as they will frequently be referred to. CONIC SECTIONS. 33 Since SP: PM:: SA: AX, and SA: AX:: AA': XX', (Prop. II.)... SP: PM:: AA': XX', So S'P: PM':: AA': XX',.'. SP - S'P: PM + PM':: AA': XX'. But PM + PM'= JlIM' = XX',. SP +S'P = AA'. COR. 1. By means of this property the ellipse may be practically described and the form of the curve determined. Let a string, equal in length to AA', have its ends fastened to two points S and S'; and let it be kept stretched by means of the point of a pencil at P; then since SP + S'P will be always equal to AA', the point P will trace out the ellipse. CoR. 2. The line AA' is the longest line that can be drawn in the ellipse.; For, if any other line PQ be drawn, then SP+ SQ > PQ, and S'P + S'Q > PQ,.. SP +SP + SQ +S' Q>2PQ, or AA' > PQ. 25. DEF. If BCB' be drawn at right angles to ACA', meeting the ellipse in B and B', it will be seen further on (Prop. XIII. Cor. 2) that BCB' is the shortest chord that can be drawn through the centre of the ellipse. (See fqg. Prop. IV.) I) CONIC SECTIONS. AA' is called the Major Axis and BB' the Minor Axis of the ellipse. In most geometrical treatises the ellipse is defined as the curve traced out by a point which moves in such a manner that the sum of its distances from two fixed points is always the same; but it appears that the properties of the curve are more clearly exhibited by defining it in a manner analogous to the parabola, and deducing immediately from that definition the property in question. Having now shown that one definition necessarily includes the other, we are at liberty in our future investigations to make use of whichever property is most convenient. PROP. IV. 26. If B C be the semi-minor axis of the ellipse, then B C = CA2 - CS2; and if SL be the semi-latus rectum, SL. AC= BC2. B L_~ 11-1 Join SB, S'B; then since SB + S'B = AA', (Prop. III.) and that SB = S'B,.'. SB=AC. But B C2 = SB2 - CS2,.~. BC2=CA -CS2. CONIC SECTIONS. 35 Again, SL: SX:: SA: AX, CS: CA, (Prop. II.) SL. AC = CS. SX, = CS. CX - CS2, (Euclid, II. 3,) = CA2 -CS2 (Prop. I.) BC2. PROP. V. 27. The sum of the distances of any point orom the foci of' an ellipse will be less or greater than AA' according as the point'is inside or outside the ellipse. A ~~~~~~~~~~~~~A (1.) Let 9 be a point inside the ellipse. Join SQ, 5'Q; and produce SQ to meet the ellipse in P; join S'P; then since S'P~+ P > S'Q, S'P + SP > S'Q+ SQ. But S'P + SP = AA', (Prop. Ill.) SQ + SQ' < AA'. (2.) Let Q be a point outside the ellipse. Join S Q, 5' Q, and let S Q meet the ellipse in the point P; join S'P; then since S'`Q + P > S'P, S 'Q~ SQ3 > SP P S'P, But SP + S'P = AA', (Prop. III.) SQ + S'Q > AA'. i)2 36 CONIC SECTIONS. COR. Conversely, a point will be inside or outside the ellipse, according as the sum of its distances from the foci is ess or greater than AA'. 28. DEF. If a point P' be taken on the ellipse near to P, (see fig. Prop. VI.) and PP' be joined, the line PP' produced, in the limiting position which it assumes when P' is made to approach indefinitely near to P, is called the Tangent to the ellipse at the point P. PROP. VI. If the tangent to the ellipse at any point P intersect the directrix in the point Z, and if S be the focus corresponding to the directrix on which Z is situated, then SZ will be at right angles to SP. Zi Let P' be a point on the ellipse near to P. Draw the chord PP', and produce it to meet the directrix in Z; join SZ. Draw PM, P'M' at right angles to the directrix, and join SP, SP'. CONIC SECTIONS. 37 Produce PS to meet the ellipse in the point Q; then since the triangles ZMiiP, ZM'P' are similar,..ZP: ZP':: P: M'P',:: SP: SP'. SZ bisects the angle P SQ. (Euclid, VI. Prop. A.) Now when P' is indefinitely near to P, and PP' becomes the tangent at the point P, the angle PSP' becomes indefinitely small, while the angle QSP' approaches two right angles; and therefore the angle ZSP' being half of the angle P' S Q, becomes ultimately a right angle. Hence when PZ becomes the tangent at the point P, the angle ZSP is a right angle, or SZis perpendicular to SP. Con. 1. Conversely, if SZ be drawn at right angles to SP, meeting the directrix in Z, and PZ be joined, PZ will be the tangent at P. CoR. 2. If ZP be produced to meet the other directrix on the point Z' and S'Z' be joined, then S'Z' is at right angles to S'P. CoR. 3. The tangents at the extremities of the latus rectum or double ordinate through the focus meet the axis produced in the point X. ProP. VII. The tangent to the ellipse at any point P makes equal angles with the focal distances SP and S'P. Let the tangent at P meet the directrices in Z and Z'. Draw MPM' at right angles to the directrices, meeting them in M and M' respectively; join SZ, S'Z'; then SP: PM:: S'P: PM'; and since the triangles MLPZ, M'PZ', are similar, PM: PZ:: PM': PZ',. SP: PZ:: S'P: PZ'. (Ex cequali.) 38 CONIC SECTIONS. Now in the triangles SPZ, S'PZ', because the sides about the angles SPZ, S'PZ' are proportional, and the angles PSZ, PS'Z' are equal, being right angles, and the angles SZP, S'Z'P are each less than a right angle,.. the triangles SPZ and S'PZ' are similar, (Euclid, VI. 7).'. the angle SPZ =the angle S'PZ'. COR. If S'P be produced to W; then the angle SPZ = the angle WPZ. PROP. VIII. The tangents at the extremities of a focal chord intersect in the directrix. Let PSQ be a focal chord, and let the tangent at P meet the directrix in Z. Join SZ; then CONIC SECTIONS. 39 the angle ZSP is a right angle, (Prop. VI.) and.. also the angle ZSQ is a right angle, Z Q is the tangent at Q; (Prop. VI. Cor. 1) or the tangents at the extremities of a focal chord intersect in the dirdctrix. PROP. IX. 29. If the tangent at P meet the axis major produced in T, and PN be the ordinate of the point P, then CT. CN.= CA2. T Xi A SN (I S IA'KV' IDraw Mi1I' parallel to the axis major meeting the directrices in ill and f'; and produce S'P to W; then, since PT bisects the angle SP W, (Prop. VII. Cor.) S'T: ST:: S'P: SP, (Euclcd, VI. A.) PM/': PM, X'N: XNY, S'T + S 7T: S' T - -S T:: X'NV + XNY: XNV - XN; or2CT: 2 CS:: 2CX: 2 CIV or CT: CS:: CX: CIV CT. CN= CS. CX, CA. (Prop II.) 40 CONIC SECTIONS. PROP. X. If on the major axis of an ellipse as diameter a circle be described and a common ordinate NPQ be drawn meeting the ellipse in P and the circle in Q, then the tangents to the ellipse and circle respectively at the points P and Q will meet the major axis produced in the same point. 't A 2V C A' Let the tangent to the ellipse at P meet the major axis produced in T; join C Q, Q T; then, by the last Proposition, CT. CIV= CA= CQ2,. the angle C Q T is a right angle. And therefore Q T is the tangent to the circle at Q; or the tangents at P and Q meet the major axis produced in the same point T. The circle described on AA' as diameter is called the Auxiliary Circle on account of the important aid that it affords in investigating the properties of the ellipse. 30. DEF. The line PG, drawn at right angles to the tangent PT, is called the Normal to the ellipse at the point P. PROP. XI. If the normal at P meet the major axis in the point G; then SG: SP: CS: CA, CONIC SECTIONS. 41 T X A SNGA 8 A'X' Since PG is at right angles to TPt, *. the angle GPT= the angle GPt. But the angle SPT= the angle S'i t, (Prop. VII.) the angle SPG = the angle B'PG, or PG bisects the angle SPS', CG B'G:: SP: S"P, (Eucliid, VI. 3.) SG: G + S'G::SP:SP+S'P; or BG: SB':: BP: AA'; or SG SP S:BB':AA'; orGS: SP:: GB: CA. Com. Hence also, B'G: S'P:: CS CA. PROP. XII. 31. If the normal at P meet the major axis in C, and PN be the ordinate at the point P, then (see fig. Prop. XI.) NG: NC:: BC2': AC'. Draw MVPM' parallel to the axis meeting the directrices in M and 11'; join BP, B'P; then, since PG bisects the angle BPS', (Prop. XL.) S'G::G S'P: SP, PM': PlM, X' N: XiV\ B'G - SG:S'G + SG::X'N - XN:X'.v + XN; or 2 CG: SS' 2 CN: XX'. 42 CONIC SECTIONS. Alternately, 2 C G: 2 ON SS': XX'; or OG: ON:: Cs Cx, CS'2 CA', (Prolp. II. Cor.) N - C G ON CA- 0C2: CA2; orG: O::BC2: A C2. PR'OP. XIII. 32. If PN be the ordinate of any point P on the ellipse then PN: ANY. A'N:: BC2: AC2. 1T A N'O A' Produce NP to meet the auxiliary circle in the point Q, and draw the tangents P 7' Q T meeting the major axis produced in the point T. (Prop. X.) Join C Q, and let the normal at P meet the ellipse in G; then, by the last Proposition, NCG: ONT:: BC2: A C2. And rectangles of the same altitude are to another as their bases,.. TN. NAG: TN. CNlT:: BC 2:A C2; or PNy, 2: QNX:: B 02: A 02. (Eyuclid, VI. 8, (or.) But QN2 = ANX. A'N, since the angle A QA' in a semicircle is a right angle,. _.PN 2:ANA'N::B2: A C2. Coi. 1. Also PN: QN::BC: A C. This result is the basis of many of the future Propositions of the ellipse. CONIC SECTIONS. 43 COR. 2. Since PA2: QN2:: BC2: AC2,. PNT2: AC2 - CN2:: BC2: AC2,.. PN2: AC - CN2 - PN2:: BC2: AC2 - BC2, or PN2: AC2 - CP2:: BC2: AC2 - BC2. Now PN2 is always less than B C2,. P. 2 is greater than BC2, *. BC is the shortest line that can be drawn to the ellipse from the centre. PtOP. XIV. If the tangent at any point P of an ellipse meet the minor axis CB produced in t, and Pn be drawn at right angles to CB; then Ct Cn- BC2 / " 2 A JV ' A' Draw the common ordinate IVPQ to the ellipse and the auxiliary circle; and let the tangents at P and Q to the ellipse and circle respectively meet the major axis produced in T (Prop. X.) and the minor axis produced in t and U. Join C Q meeting Pn in R; then since PR is parallel to CN, CB: CQ::PN: Q,:: BC: AC. (Prop. XIII. Cor. 1.) 44 CONIC SECTIONS. But CQ = AC,. CR = C. (1) Again, joining Et, Ct: CU:: PN: QN,:: CR: CQ,. it is parallel to QU,.*. Cl(7 is a right angle,.. Ct CRn = (Euclid, VI. 8, Cor.) But CR= BC, (1) Ct. Cn = BC. This proposition also admits of a demonstration similar to that given for the corresponding property of the hyperbola. PROP. XV. 33. If from the foci S and S', SY, and S' Y' are drawn at right angles to the tangent at P, then Y and Y' are on the circumference of the auxiliary circle, and SY. S'Y' = BC. IA' CONIC;ECTIONS. 45 Join SP, S'P, and produce SY and S'P to meet in W; and join CY; then since the angle SPY = the angle WP Y, (Prop. VII. Gor.) and the angle S YP = the angle I YP, and the side P Y is common to the triangles SPY, WP Y, the triangle SPY = the triangle TVP Yin all respects, SP= PlY, SP+ S'P = S' T But SP + S P = AA', (Pr~'p. IIIT.) SI'W=. AA'. Again SC C=S'CandSY = YVW,. SC: S'C:: SYZ~: Y~v'.'. CYis parallel ~to S'W,. Y: S'W:: CS: SS, CY= 'S'W= CA. So CY' = CA, Y and Y' are points on the auxiliary circle. Next let YS be produced to meet the auxilliary circle in Z, and join Z Y'; then since the angle Z YY' is a right angle, Z Y' passes through the centre C, the angle SCZ the angle S'CY'. SZ = Y. SY. S. Y, S6YY. SZ, =AS. A'S, (Euclid, IMI. 35) = CA - CS', (Euclid, II. 5) = BC2. (Prop. IV.) COR. If CD be drawn parallel to the tangent at P, meeting SP in E; then 4G CONIC SECTIONS. since the figure C YPE is a parallelogram,.'. PE= CY= AC. PROP. XVI. 34. To draw a pair ternal point 0. ^^^ of tangents to an ellipse from an cx lrith centre S' and radius equal to AA' describe a circle. Join OS, OS'; and let SO or 8'0 produced meet the circle in the point I. Now, if 0 be a point outside the circle MIM', it is evident that OS is greater than OI; and if 0 be inside the circle, since OS + OS' > AA' or S'I, (Prop. V.). OS> O. With centre 0 and radius OS describe another circle cutting the former in the points M and M', which it will always do since OS is greater than OL CONIC SECTIONS. 47 Join S'M, S'M', meeting the ellipse in the points P and P'. Join OP, OP'; these will be the tangents required. Join SP, SP'; then, since SP + S'P = AA' = S'M,.. SP= Pi. And.' SP, PO = MP, PO each to each, and OS= O,.. the angle OPS = the angle OPM,.'. OP is the tangent at P. (Prop. VII. Cor.) So OP' is the tangent at P'. PROP. XVII. If from a point 0 a pair of tangents OP, OP' be drawn to an ellipse, then OP and OP' will subtend equal angles at either focus. Join SP, S'P; SP', S'P'; and produce S'P, S'P' to J and H', making Pi equal to SP, and P'M' equal to SP. Join O, OM'; OS, OS'. Then since OP, PS = OP, PM, each to each, and the angle OPS = the angle OPM, (Prop. VII. Cor.)... os= o and the angle OSP = the angle OMP. So OS= OM', and the angle OSP' = the angle OM'P',.O. OM= M0'. Again, *. S'M = S'P + SP = AA', and S'M' = SP' + SP' = AA',.. 'M= S'M', And., OS', S'M = OS', S'M', each to each, and OM= OM',.. the angle OS'M = the angle OS'M', and the angle OMS' = the angle OM'S'. 48 CONIC SECTIONS. But the angle OMS' = the angle OSP, and the angle OMl'S' = the angle OSP',.'. the angle OSP = the angle OSP',. OP and OP' subtend equal angles at either focus. PnoP. XVIII. 35. If from an external point 0 a pair of tangents OQ, OQ' be drawn to an ellipse, and CO be joined meeting the chord QQ' in V, and the ellipse in P; then (1.) QQ' will be bisected in V. (2.) The tangent at P will be parallel to QQ' (3.) C.P will be a mean proportional between CV and CO. Produce O Q, OQ' to meet the major axis produced in T and T. Draw the ordinates NQ, N' Q', and produce them to meet the circle in q and q'. CONIC SECTIONS. 49 Then Tq and T'q' will be tangents to the auxiliary circle. (Prop. X.) Let Tq and T'q' be produced to meet in o; join Co meeting the chord qq/ in v, and the circle in p. Now, since the corresponding ordinates of the ellipse and auxiliary circle are in the constant ratio of BC to AC, the three lines ol, pm, vn drawn at right angles to AA' will pass through the points 0, P, V respectively. For, according as 0 is the point where o meets TQ or T'Q' we shall have 10: lo:: N7Q:q,:: BC: AC; or 10: lo:: N'Q': N'',::BC: AC, '. Oo is perpendicular to A A'. So Pp and Vv are perpendicular to A A', o. Pp, Vv are parallel. Hence (1.) QV: VQ':: v: v'. But qv = vg' from the circle,.. QT=VQ'; or Q Q' is bisected in V. (2.) Since NQ: Nq:: N'Q': N'q' it is evident that QQ' and qq' will meet the axis produced in the same point. Also the tangents to the ellipse and circle at P and p respectively will meet the axis in the same point. Now in the circle the tangent at p is manifestly parallel to qq', and NQ: Nq:: mP: mp,. the tangent at P is parallel to QQ'. (3.) If Cq be joined, since the angle Cqo is a right angle and Co is perpendicular to gq',. Cv: Cq:: Cq: Co, (Euclid, VI. 8 Cor.) or, since Cq = Cp, Cv: Cp:: Cp: Co. E 50 CONIC SECTIONS. But Cv: Cp:: CV: CP, and C: Co:: CP: CO,.CV: CP:: CP: CO, CO. C V= CP2. Co'. From this it is manifest that if any number of chords be drawn parallel to each other in an ellipse, their middle points will all lie on the line drawn fiom the centre to the point where the tangent parallel to the chord meets the ellipse. DEF. The line PCP' drawn through the centre of an ellipse and meeting the curve in P and P', is called a Diameter. The diameter consequently bisects all chords parallel to the tangents at its extremities; and the tangents at the extremities of any chord will intersect the diameter corresponding to that chord in the same point. 36. DEF. If CD be drawn parallel to the tangent at P, then CD is said to be conjugate to CP. PROP. XIX. In the ellipse if CD be conjugate to CP, then will CP be conjugate to CD. Draw the ordinates PN, DR, and produce them to meet the auxiliary circle in the points p, d. Join CP, Cp; CD, Cd; and draw the tangents TP, Tp; T'D, T'd. Now, since CD is parallel to PT, CONIC SECTIONS. 51.. the triangle PNT is similar to the triangle DR C... TN: CR::PN: DR,:: Np: Rd, (Prop. XIII. Cot.).. Tp is parallel to Cd,.. the angle p Cd is a right angle,.'. Cp is parallel to T'd,.. the triangle p CN is similar to the triangle d T'R,.. NC: T':: Np: Rd,:: NP: RD,.O. CP is parallel to DT', C. CP is conjugate to CD. COR. Since CRd and CNp are each similar to dl-T', Euclid, VI. 8).. the triangle CRd is similar to the triangle C2)p, and the side Cd = the side Cp,.. the triangle CRd = the triangle COAp in all respects,.'. CN = Rd, and CR = Np. Hence DR: COT:: DR:Rd,:: BC: A C; also PN: CR:: PN: Np,:: BC: AC. PRoP. XX. 37. If CP and CD be conjugate semi-diameters, and PX, DR be the ordinates of the points P and D; then (1.) CN + G~C = AC2. (2.) PN12 + D R = B C2. (3.) CP2 + CD2 = A C2 + B C2 2 CONIC SECTIONS. Produce NP, RD to meet the auxiliary circle in the points p, d then ON = Bd, (Pr-op. XIX. Cor.) CN2~+CR2 =Bd' + COK, = Cd2, = CA2. Again, PN: Np B C A C, P2 Ap2 BC2 AC2. So D R2 Rd2 B C2 A C2.P 2DB2:Npj+ Rdl:: C2:A 2 but NYP2 + Rd 2 = CR2 + ON 2 = A 02, PNV2+ DR2== B C2 and ONV2 + CR2 = A4 02, CP2~+CD 2=A C2~+BC2 38. DEF. A line Q V drawn parallel to the tangent at P, and meeting GP in V7, is called an Ordinate to the diameter OP. PROP. XXI. If Q V be any ordinate to the dliameter.P CP', and CD be coujugate to CP; then QV2 PV. P'V::CD2:O2 CONIC SECTIONS. 5 53 Draw the tangent UQ W meeting OP and CD produced in U and 11; and draw 911 parallel to CP, meeting CD in,-R. Now, since CR or QV-2 U:CD G: D:C W, (Prop. XVIII.) C2:: CR: CW' C2:: UV: CU (Euclid, VI. 20 Cor.) 11" 'It Again, )(rp VI. since C U: CP:: CP: C 1,(rp VI...CU: 017:: OP2: 0172,. (Euclid, VI. 20 Cor.) _.CUCVTCU::GCpCV2: CP2, or UV: CU:: PV. P'V: OP2. Hence Q V2 or QV2: PV~. CD 2:: P17.P'V: OP2. P'V:: C'D2: OP2. PR'OP. XXII. 39. The area of any parallelogram formed by drawing tangents to an ellipse at the extremities of a pair of conjugate 0-4 54 ~~~~CONIC SECTIONS. diameters is equal to the rectangle contained by the axes of the ellipse. Let POP', DOD' be a pair of conjugate diameters, and let a parallelogram be formed by drawing tangents at the points P, P' D, D'. Let the tangent at P melet CA produced in T; join D'T~ Draw the ordinates PN, DR?, Dift; then since PT is parallel to CD', the parallelogram PD' is double the triangle C TD', and therefore equal to the rectangle contained by C T and D'B. Now lif': ON:: B C: AOC, (Prop. XIX. C'or.) C T. URf' C CT. ON:: B C: AC (Euclid~ VI. 1.) BC. AC: A4C2. (EUClid, VI. 1.) But CT. CN-. AC',. CT. D'B'AC. BO, the parallelogram LL' 4 the parallelogram PD', ==4 A C. B 0, ==AA'. BB. CONIC SECTIONS. Con. If PF be drawn at right angles to D CD' meeting CD' in F; then PF. CD' = area of parallelogram PD', = AC. BC. PROP. XXIII. 40. If CP and CD be conjugate diameters, and PF be drawn at right angles to CD meeting CA in C, then PF. PG = BC2. Draw the ordinate PN, and produce it to meet CD' in K. Also draw Pn at right angles to CB, and let the tangent at P meet CB produced in t. Now since the angles at N and F are right angles, it is evident that a circle may be described about the quadrilateral figure NKFG; PG. PF= PN. PFPK, (Euclid, III. 36 Cor.) = Ct. Cn, = BC2. (Prop. XIV.) 56 CONIC SECTIONS. PROP. XXIV. 41. If P be any point on the ellipse, and CD be conjugate to CP, then SP. S'P= CD2. / / Draw the normal PG and produce it to meet CD' in F; then since CD' is parallel to the tangent at P,. PF is at right angles to CD',.. PF. CD = AC. BC, (Prop. XXII. Cor.) and PF. PG = BC2 = BC. BC, (Prop. XXIII.)... CD PG:: AC: BC. (1) Again, SP: S:: CA: CS, (Prop. XI.) S'P: 'G:: CA: CS. (Prop. X.) Compounding SP. S'P: SG. S'G:: CA': CS2,. P. P. 'P: SP. S'P- SG. S'G:: CA2: CA' - CS2. But SP. 'P - SG. S'G = PG2, (Euclid, VI. Prop. B).. P. S'P: PG2:: CA2: BC2. But from (1) CD2: PG2:: CA2: BC2,.. SP. S'P = CD2. This proposition may also be very easily deduced from Prop. XV. CONIC SECTIONS. 57 PROP. XXV. 42. The area of the ellipse is to the area of the auxiliary circle as BC to AC. P B A N' C A' Let PN and P'N' be two ordinates of the ellipse near together. Produce NP, N'P', to meet the auxiliary circle in Q and Q'. Draw Pm, Qn, perpendicular to Q'N'. Then the parallelogram PN': the parallelogram QN': PN: Q',, BC:: AC. And the same will be true for all the parallelograms that can be similarly described in the ellipse and auxiliary circle. Hence the sum of all the parallelograms inscribed in the ellipse is to the sum of all the parallelograms inscribed in the circle as B C to A C. And this holds however the number of parallelograms be increased. But when the number of parallelograms is increased, and the breadth of each diminished indefinitely, the sum of the parallelograms inscribed in the ellipse will be equal to the area of the ellipse, and the sum of those inscribed in the circle to the area of the circle. Hence the area of the ellipse: the area of the circle:: BC: AC. 58 CONIC SECTIONS. 43. DEF. If with a point 0 on the normal at P as centre, and OP as radius, a circle be described touching the ellipse at P, and cutting it in Q; then, when the point Q is made to approach indefinitely near to P, the circle is called the Circle of Curvature at the point P. PROP. XXVI. If PH be the chord of the circle of curvature at the point P of an ellipse, which passes through the centre; then PH. CP= 2 CD2. Let PT be the tangent, and PG the normal at the point P. D P u With centre 0, and radius OP, describe ellipse in the point Q. a circle cutting the Draw R QW parallel to CP, meeting the circle in W, and TP produced in R. CONIC SECTIONS. 59 Also draw QV parallel to PBR, meeting the diameter PP' in V; -then since PTP touches the circle at P, BQ. BIy= PB2, (Euclid, III. 36) or PV.BTW= -QV2. But QV2: PV. P'V:: CD2C: GP, (Prop. XXI.) PV. R BTV: PV. P'V:: CD2: CP2, orBR W: P'V:: GD2': OP2. Now, when the circle becomes the circle of curvature at P, the points B and Q move up to, and coincide with P, and the lines BRW and PH become equal, while P V becomes equal to PP', or 2 OP. Hence, P11: 2 OP:: GD2: CP'2 PH. CP: 2CP2::2D2: 2CP2, PH. PC 2 CD2. PROP. XXVII. If PU be the diameter of the circle of curvature at the point P of the ellipse, and PE be drawn at right angles to CD; then P U. P -' = 20CD2 Since the triangle PHU is similar to the triangle PGC, PUU: PH:: CP:PF, PU. PF= =PH. CP, = 2 CD2. (Prop. XXVI.) PROP. XXVIII. If PI be the chord of the circle of curvature through the focus of the ellipse; then PI. AG-= GD2 CONIC SECTIONS. Let PI meet CD in E; then, since the triangles PIU and PEF are similar,.'. P PU:: PF: PE. But PE = A C, (Prop. XV. Cor.).. P: PU:: PF: AC,. PI. A C = PU. PF, = 2 CD2. (Prop. XXVII.) PROP. XXIX. 44. If two chords of an ellipse intersect one another, the rectangles contained by their segments are proportional to the squares of the diameters parallel to them. Let POP' be any chord drawn through the point 0, and let CD be the semi-diameter parallel to it. Draw the ordinates NP, N'.P', MD, and produce them to meet the auxiliary circle in Q, Q', D'; then since NP: NVQ:: N'P': N'Q', (Prop. XIII. Cor.) it is evident that PP' and Q Q' will meet in the same point T. the axis produced Q' Also since NP: NQ:: MD: MRD', (Prop. XIII. Cor.) and TPP' is parallel to CD,.. TQQ' is parallel to CD'. CONIC SECTIONS. 61 Draw EO parallel to NQ or N' Q', and produce it to meet QQ' in 0'; then PO: QO':: TO: TO', and P'O: Q'O':: TO: TO',.. PO. PO: Q O'. Q'O':: TO2: TO',:: CD2: CD'",:CD2: A C2. Alternately, PO. P'O: CD':: QO'. Q'O': AC2. Again, if through the point 0 any other chord pOp' be drawn, since EO: EO':: BC: AC, it is manifest that the corresponding chord qq' in the auxiliary circle will pass through the point 0'; and if Cd be the semidiameter parallel to pp' we shall have as before, pO. p' O: Cd2:: qO'. q'O': A C2. But Q O'. Q'O' = q O'. ' 0', (Euclid, III. 35).. PO. P'O: CD':: pO.p'O: Cd2. or PO. P'O: pO.20:: CD': Cd2. The same result may be shown to be true when the point 0 is without the ellipse. PuoP. XXX. If QVQ' be any ordinate to the diameter CP, the circle described through the three points P, Q, Q' will intersect the ellipse in a fourth point, which depends only upon the position of P. Draw the ordinate PN, and produce it to meet the ellipse in P'; then, since, if VNT be the subtangent of either P or P', CT. CN= A C2, (Prop. IX.) therefore the tangents at P and P' will meet the major axis produced in the same point T. CONIC SECTIONS. Draw PR parallel to TP', meeting the ellipse in R, and Q Q' in 0; then if CD and CD' be drawn parallel respectively to TP and TP', meeting the ellipse in D and D', PO. OR: QO. OQ':: CD'2: CD2. (Prop. XXIX.) But CD' = CD, since CP'= CP, P.. O OR= QO. OQ'. Hence, by the converse of Euclid III. Prop. 22, the point R is on the circle which passes through P, Q, Q'. COR. When the point V is brought indefinitely near to P, QQ' coincides with the tangent to the ellipse at P, and becomes also a tangent to the circle at P since Q and Q' are indefinitely near to each other. The circle therefore becomes the circle of curvature at the point P. Hence, if PR be drawn parallel to the tangent at P', or be equally inclined to the axis with PT, it will meet the ellipse in the point where the circle of curvature at P intersects the ellipse. PROBLEMS ON THE ELLIPSE. 1. IN what position of P is the angle SPS' greatest? 2. The latus rectum is a third proportional to the axis major and axis minor. 3. Construct on the axis minor as base, a rectangle which shall be to the triangle SLS' in the duplicate ratio of the major axis to the minor axis, L being the extremity of the latus rectum. 4. If a series of ellipses be described having the same major axis; the tangents at the extremities of their latera recta will all meet the minor axis in the same point. 5. Find the locus of the centres of all the ellipses having the same focus, and their major axes of the same length, and touching a given straight line. 6. Given the foci, it is required to describe an ellipse touching a given straight line. 7. If PT be a tangent to an ellipse, meeting the axis in T, and AP, A'P, be produced to meet the perpendicular to the major axis through T in Q and Q', then QT = Q'T. 8. If the angle SBS' be a right angle, prove that CA2= 2CB. 9. If CP be a semi-diameter, and A Q 0 be drawn parallel to CP meeting the curve in Q, and CB produced in 0, then C2 = AO. AQ. 10. If AB, CD, which are not parallel, make equal angles with either axis, the lines AC, BD, as also AD, BC, will make equal angles with either axis. 64 CONIC SECTIONS. 11. PSp is any focal chord. PA and pA are produced to meet the directrix in Q and q. Prove that the angle QSq is a right angle. 12. If a circle be described touching the axis major in one focus, and passing through one extremity of the axis minor; A C will be a mean proportional between the diameter of this circle and B C. 13. If PQQ'P' be a chord of the auxiliary circle, and a circle be described on the minor axis as diameter, cutting the chord in Q and Q', then PQ. P'Q = CS2. 14. If PG be the normal at P, and GL be drawn at right angles to SP, then PL = ~ latus rectum. 15. The sum of the squares of the normals at the extremities of conjugate diameters is constant. 16. If on the normal at P, PQ be taken equal to the semiconjugate diameter CD, the locus of Q is a circle whose radius is AC - BC. 17. Find the locus of the intersection of a pair of tangents at right angles to each other. 18. P is any point on an ellipse. To any point Q on the curve draw AQ, A'Q, meeting NP in R and S, and prove that NR. NS = NP2. 19. If PG be a normal, and GL perpendicular to SP, the ratio of GL to PN is constant. 20. If NP produced meet the tangent at the extremity of the latus rectum in Q, then QN= PS. 21. In an ellipse the tangent at any point makes a greater angle with the focal distance than with the perpendicular on the directrix. 22. A diameter of an ellipse, parallel to the tangent at any point, meets the focal distances of the point, and from the points of intersection lines are drawn perpendicular to the focal distances. Prove that these lines intersect in the axis minor. 23. The subnormal is a third proportional to CT and B C, CONIC SECTIONS. 65 24. If PN be the ordinate of P, prove that NY: NY' P Y: PY'. (See fig. Prop. XV.) 25. If from C lines be drawn parallel and perpendicular to the tangent at P, they inclose a part of one of the focal distances of that point equal to the other. 26. If P be a fixed point on an ellipse, and QQ' an ordinate to CP, the circle QPQ' will meet the ellipse in a fixed point. 27. P is any point on an ellipse. Draw PP' parallel to the axis major, and through P' draw P'Q, P'Q', making equal angles with the major axis. Join QQ'; then Q Q' is parallel to the tangent at P. 28. What parallelogram circumscribing an ellipse has the least area? 29. When is the square of the sum of conjugate diameters least? 30. Given the axes of an ellipse, and the position of one focus, and of one point in the curve, give a geometrical construction for finding the centre. 31. If lines drawn through any point of an ellipse to the extremities of any diameter meet the conjugate CD in -M and NV, then CM. CI = CD2. 32. If CP and CD be conjugate, prove that (SP - AC) + (SD - AC)2 = SC2 33. If CP and CD be conjugate, and BP, BD be joined, as also AD, A'P, these latter meeting in 0, then BD OP is a parallelogram. When is the area greatest? 34. If PSp, Q Cq be two parallel chords through the focus and centre of an ellipse, prove that SP. Sp: CQ. Cq:: BC2: AC. 35. If the tangent at the vertex A cut any two conjugate diameters in T and t, then A T. A t = B C2. 36. If the tangents at three points P, Q, R, intersect in R,, Q,, P, prove that PR,. PQ. QR = PQ,. R,Q. PR. F 66 CONIC SECTIONS. 37. If a circle be described touching SP, S'P produced, and the major axis of the ellipse, find the locus of the centre. 38. If from the extremities of the axes of an ellipse any four parallel lines be drawn, the points in which they cut the curve are the extremities of conjugate diameters. 39. If two equal and similar ellipses have a common centre, the points of intersection are at the extremities of diameters at right angles to one another. 40. If PSQ be a focal chord, and X the foot of the directrix, XP and XQ are equally inclined to the axis. 41. 0P, 0 Q are tangents to an ellipse, and P Q is produced to meet the directrices in R, B', prove that RP.R'P: RQ: R'Q:: OP2: OQ2. 42. NPQ is a common ordinate to the ellipse and auxiliary circle. PR, QR are normals at P and Q intersecting in R. The locus of R is a circle whose radius is A C + B C. 43. If the conjugate to CP meet SP, S'P, or these produced in E, E'; then SE= S'E', and the circles circumscribing S CE, S' CE' are equal. 44. The locus of the middle points of all focal chords in an ellipse is a similar ellipse. 45. The circle described about the triangle SBS' will cut the minor axis in the centre of the circle of curvature at B. 46. The locus of the centre of the circle inscribed in the triangle SPS' is an ellipse. 47. If a circle be described intersecting an ellipse in four points, and chords be drawn through the points of intersection, diameters parallel to the chords will be equal. 48. An ellipse slides between two lines at right angles to each other, find the locus of its centre. 49. If from the focus S perpendiculars be drawn upon the conjugate diameters CP, CD, these perpendiculars produced backward will intersect CD and CP in the directrix. 50. Find the point at which the diameter of curvature is a mean proportional between the major and minor axes. CONIC SECTIONS. 67 51. The circle of curvature at a point, where the conjugate diameters are equal, meets the ellipse again at the extremity of the diameter. 52. The locus of the intersection of lines drawn from A, A' at right angles to AP, A'P is an ellipse. 53. If a quadrilateral figure be inscribable in two ellipses whose major axes are parallel or perpendicular, any two of its opposite angles will be equal to two right angles. 54. If CIV, NP are the abscissa and ordinate of a point P on a circle whose centre is C, and N Q be taken equal to NlP, and be inclined to it at a constant angle, the locus of Q is an ellipse. 55. If two ellipses having the same major axes can be inscribed in a parallelogram, the foci will be on the corners of an equiangular parallelogram. 56. Two ellipses, whose major axes are equal, have a common focus. Prove that they intersect in two points only. 57. A circle described about the triangle SPS' cuts the minor axes in R on the opposite side to P. Prove that SR varies as the normal PG. 58. If r and R be the radii of the circles inscribed in and about the triangle SPS, prove that R. r varies as SP. S'P. 59. The circle described upon PG as diameter cuts SP, S'P in K and L. Prove that KL is bisected by PG, and is perpendicular to it. 60. If from S' a line be drawn parallel to SP, it will meet S Y in the circumference of a circle. 61. T and t are the points where the tangent at P meets the axes. CP is produced to meet in L the circle described about the triangle TCt; prove that PL is half the chord of the circle of curvature at P in the direction of C, and that CP. CL is constant. 62. About the triangle PQ R an ellipse is described, having its centre at the point where the lines drawn from P, Q, R to the middle points of the opposite sides meet. CP, CQ, CR are produced to meet the ellipse in P', Q', a'. Prove that F 2 68 CONIC SECTIONS. the tangents at P', Q', R' form a triangle similar to P QR, and four times as large. 63. Lines from Y and Y' perpendicular to the major axis cut the circles on SP, S'P as diameters in I and J. Prove that IS and JS' when produced, intersect B C in the same point. 64. If from the ends of any diameter chords be drawn to any point in the ellipse, the diameters parallel to these chords will be conjugate. 65. If T be the angle between tangents at the extremities of a focal chord, and 0 the angle subtended by the chord at the other focus, then 2 T + O = 2 right angles. 66. The acute angles which SP, S Q make with the tangents are complementary. Prove that B C2 is a mean proportional between the areas of the triangles SPS', S QS'. Also, show that the problem is impossible unless B C < CS. 67. A series of ellipses have their equal conjugate diameters of the same magnitude. One of these diameters is fixed and common, while the other varies. The tangents drawn from any point in the fixed diameter produced will touch the ellipses in points situated on a circle. 68. If on the longer side of a rectangle as major axis an ellipse be described, passing through the intersection of the diagonals, and lines be drawn from any point of the ellipse exterior to the rectangle to the ends of the remote side, they will divide the major axis into segments, which are in geometric progression. 69. From any point P of an ellipse PQ is drawn at right angles to SPmeeting the diameter conjugate to CP in Q. Prove that PQ varies inversely as the perpendicular from P on the major axis. 70. In an ellipse SQ and S'Q, drawn at right angles to a pair of conjugate diameters, intersect in Q. Prove that the locus of Q is a concentric ellipse. CHAPTER III. THE HYPERBOLA. 45. DEF. The Iyperbola is the curve traced out by a point which moves in such a manner that its distance from a given fixed point continually bears the same ratio, greater than unity, to its distance from a given fixed line. (See Introduction.) PROP. I. The focus and directrix of a hyperbola being given, to find any number of points on the curve. Let S be the focus, and MIX the directrix. Draw SX at right angles to the directrix, and divide SX in the point A, so that SA may be to AX in the given fixed ratio, greater than unity; then A is a point on the curve. On SX produced take a point A', such that SA': A'X:: SA: AX; then A' will also be a point on the curve. On the directrix take any point M; and through S and M draw the line SYMY', meeting AY and A'Y', drawn at right angles to AA', in the points Y and Y'; On YY' as diameter describe a circle, and draw PMP parallel to A A', cutting the circle in the points P and P'; P and P' will be points on the hyperbola. 70 CONIC SECTIONS. Join P Y, P Y', SP; then since SY: Y-21:: SA: AX, (Euclid, VI. 2) and SY': Y'-M:: SA': A'X, (Euclid, VI. 2)..SY: YM:: SY': Y'M; or, alternately, SY: SY':: YM: Y'M, and the angle YP Y' in a semicircle is a right angle, PY bisects the angle SPM,' SP: PII:: SY: YJII iSA AX. So we may show that SP': P'Fm:: SY: Y'-31i * For, if not, make the angle YPm equal to YPS; then S Ym:: SP: Pmn. (Euclid, VI. 3.) and since, if PY bisect SPm, PY' being at right angles to P Y, also bisects the angle between MP and SP produced; *SY' Y'm SP: Pm, (Eeclid, VI. A.) Hence SY: Sn', S Y'Ym, or S Y: S Y Ynm: FM, the points Hand m coincide. CONIC SECTIONS. 71:: SA: AX,..P and P' are points on the curve. In the same way, by taking other points on the directrix, we may obtain as many more points on the curve as we please. COR. 1. Since, corresponding to every point P on the curve, there is a point P' situated in precisely the same manner with respect to A' Y' as P is with respect to A Y, it is clear that if we make A'S' equal to A S, and A'X' equal to AX, and draw X'M' at right angles to AX', the curve could be equally well described with S' as focus and M'X' as directrix. The hyperbola is therefore symmetrical, not only with respect to the line AA', but also with respect to the line OC drawn through the middle point of YY' at right angles to and bisecting AA'. Con. 2. The line OP produced will bisect the angle SP W between SP and S'P produced. Produce O P and S' S to meet in G. Produce PM to meet X'M' in M', and draw OS' passing through the point ll'; then SP: PM:: S'P: PM', or, alternately, SP: S'P::: P: P'. (1) Again, SG: PM:: S'G: PM', or, alternately, SG: S'G:: PM: PM', (2).. from (1) and (2) SP: S'P:: SG: S'G,. PG bisects the angle SP I (Euclid, VI. A.) It will be shown hereafter (Prop. IX.) that the normal to the hyperbola at the point P also bisects the angle SPT. Hence the hyperbola and circle have the same tangent at the point P. The hyperbola will consequently touch all the infinite series of circles which can be described in the same manner as the one in the figure, by taking different points on the directrix. 72 CONIC SECTIONS. PROP. II. 46. If C be the middle point of AA', then CA is a mean proportional between CS and CX, or CS. CX = CA2. (See fig. Prop. III.) Si Alternate.'. Sd Aga.'. S. Alternat< Hence from nee SA': A'X ely, SA': SA '- SA: SA or AA' SA.. AA': XX' or CA: CX Lin, SA': SA I' + SA: SA or SS': SA ely, SS': AA' or CS: CA (1) and (2), CA: CX SA: AX. A'X: AX, A'X-AX: AX, XX': AX, SA: AX, SA: AX. (1.)* A'X: AX. A' + AX: AX, AA': AX. SA: AX, SA: AX. (2.):: CS: CA,.. CA2= CX. CS. Or CA is a mean proportional between CS and CX. CoR. Since the three lines CS, CA, CX, are proportional, therefore, by the definition of duplicate ratio and Euclid, VI. 20 Cor., CS: CX:: CS2: CA2. (3.) PROP. III. 47. If P be any point on the hyperbola, and S be the focus nearer to P; then S'P- SP = AA'. Since SP: PM:: SA: AX, * N.B. The results (1), (2), (3), should be remembered, as they will frequently be referred to. CONIC SECTIONS. 73 \ and SA: AX:: AA': XX', (Prop. II.).. SP: PM:: AA': XX'. So S'P: PM':: AA': XX',. S'P- SP: PM'- PM:: AA': XX'. But PM' - PM = AM' = XX',.S'P- SP = AA'. CoR. By means of this property the hyberbola may be practically described, and the form of the curve determined. Let a rigid bar S'Q of any length have one end fastened at the focus S', in such a manner that it is capable of turning freely round S' as a centre in the plane of the paper. At the other end of the bar let a string be fastened of such a length that when stretched along the bar it shall just reach to within a distance equal to AA' from the end S' of the bar. 74 CONIC SECTIONS. eC 2NIV If the loose end of the string be now fastened to the focus S, and the rod being initially placed in the position S'S, be made to revolve round S', while the string is kept constantly stretched by means of the point of a pencil at P, in contact with the bar; since S'P and SP are always increasing by the same amount, viz. the length of the portion of the string that removes itself from the bar, between any two positions of P, the difference between S'P and SP will be constantly the same, and the point P will trace out the hyperbola. Another perfectly similar branch may be described in the same manner by making the bar revolve round S as centre. In this case S'P - SP will be equal to AA'. The curve, therefore, consists of two similar branches, which recede indefinitely both from the line AA', and also from the line BOB' drawn bisecting AA' at right angles. (See fi. Prop. IV.) 48. If BC be taken of such a length that BC = CS2 - CA2, CONIC SECTIONS. 75 and CB' be made equal to CB, then AA' and BB' are called respectively the Transverse and Conjugate Axes. The line BCB' does not meet the hyperbola, and the reason of its being introduced will be seen further on. If the conjugate and transverse axes are equal, the hyperbola is said to be rectaigular or equilateral. The property of the hyperbola, which we have just investigated, viz. that the difference between SP and S'P is constant, is sometimes taken as the definition of the curve. (See Chapter II. Art. 25.) Also as in the ellipse, if SL be the semi-latus rectum, it may be proved that SL. A C= BC. PROP. IV. 49. The difference of the distances of any point from the foci of a hyperbola will be greater or less than AA', according as the point is on the concave or convex side of the curve. /-/ ~\Q/ " A ' iCU A S (1). Let Q be a point on the concave side of the hyperbola. Join SQ, S'Q, and let S'Q meet the curve in P; join SP; then since S'Q = S'P + PQ, and SQ < SP + PQ, 76 CONIC SECTIONS.. '. 'Q- SQ> S'P- SP. but S'P - SP = AA',. '. S'Q- S Q>AA'. (2.) Let Q be a point on the convex side of the curve, nearer to S than S'; join SQ, S'Q, and let SQ meet the curve in P; join S'P; then S'Q < S'P+ PQ, and SQ = SP + PQ, S. Q- SQ< 'P- SP, but S'P- SP = A',. S'Q- SQ < AA', so if Q be nearer to S' than S, we can show that SQ - S'Q < AA'; CoR. Coversely a point will be on the concave or convex side of the hyperbola, according as the difference of its distances from the foci is greater or less than AA'. 50. DEF. If a point P' be taken on the hyperbola near to P (see fig. Prop. V.) and PP' be joined, the line PP' produced, in the limiting position which it assumes when P' is made to approach indefinitely near to P, is called the Tangent to the hyperbola at the point P. PnoP. V. If the tangent to the hyperbola at any point P meet the directrix in the point Z, and if S be the focus corresponding to the directrix on which Z is situated, then SZ will be at right angles to SP. CONIC SECTIONS. 77 Let P' be a point in the curve near to P. Draw the chord PP', and produce it to meet the directrix in Z; join SZ Draw PM, P'M' at right angles to the directrix, and join SP, P'. Produce PS to meet the hyperbola in Q; then since the triangles ZMiP, ZM'P' are similar,.ZP: ZP':: lP: M'P',:: SP: SP'.. SZ bisects the angle P'S Q. (Euclid, VI. A.) Now, when P' is indefinitely near to P, and PP' becomes the tangent at the point P, the angle PSP' becomes indefinitely small, while the angle QSP' approaches two right angles; and therefore the angles ZSP', being half of the angle P'S Q, becomes ultimately a right angle. Hence, when PZ becomes the tangent at the point P, the angle ZSP is a right angle, or SZ is perpendicular to SP. 78 CONIC SECTIONS. COR. 1. Conversely, if SZ be drawn at right angles to SP, meeting the directrix in Z, and PZ be joined, PZ will be the tangent at P. COR. 2. If PZ be produced to meet the other directrix in Z', and S'Z' be joined; then S'Z' is at right angles to S'P'. COR. 3. The tangents at the extremities of the latus rectum, or double ordinate through the focus, meet the axis in the point X. PROP. VI. The tangent to the hyperbola at any point P makes equal angles with the focal distances SP and 7S'P. Let the tangent at P meet the directrices in Z and Z'. Draw PMM' at right angles to the directrices meeting them in M and M' respectively; join SZ, S'Z'; then SP: PM:: S'P: P '. CONIC SECTIONS. 79 And since the triangles ZMP, Z'M'P are similar, PM: PZ:: PM': PZ',. SP: PZ:: S'P: PZ'. (Excequali.) Now in the triangles SPZ, S'PZ' because the sides about the angles SPZ, S'PZ' are proportional, and the angles PSZ PS'Z' are equal, being right angles, and the angles SZP, S'Z'P are each less than a right angle,. the triangles SPZ, S'PZ' are similar. (Euclid, VI. 7).. the angle SPZ= S'PZ'. PROP. VII. The tangents at the extremities of a focal chord intersect in the directrix. Let PSQ be a focal chord, and let the tangent P meet the directrix in Z. Join SZ; then the angle ZSP is a right angle, (Prop. V.) And.'. also the angle ZS Q is a right angle,. ZQ is the tangent at Q. (Prop. V. Cor. 1.) Or the tangents at the extremities of a focal chord intersect in the directrix. PROP. VIII. 51. If the tangent at P meet the transverse axis in T, and PNi be the ordinate of the point P; then CT. C = CA2. Draw PMM' at right angles to the directrices meeting them in M and M'. Join SP, S'P; then since PT bisects the angle SPS', (Prop. VI.). S'T: ST:: S'P: SP, (Euclid, VI. 3.):: PM': PM,: N: X: XN. 80 CONIC SECTIONS... S'T- ST: S'T + ST:: X'N-N: X'I+ XN or 2 CT: 2S::: CX: 2CN, or CT: CS:: CX: N. CT.. CN =CS. CX, = CA2. (Prop. II.) 52. DEF. The line PG, drawn at right angles to the tangent PT, is called the Normal to the hyperbola at the point P. PROP. IX. If the normal to the hyperbola at the point P meet the transverse axis in the point G, and PN be the ordinate of the point P, then NG: NC:: BC2: A C. Draw PlMMI' at right angles to the directrices, meeting them in M and M', and produce S'P to W; then since the angle TP G is a right angle,.~ the angle WPG = the complement of the angle S'PT, and the angle SPG = the complement of the angle SP T; CONIC SECTIONS. 81 81 but the angle S'P T = thle angle SP T, the angle WR C = the angle SR G, PC bisects the angle SP W, S'C SCG S'P SP, (Eaclid, VI.. PM' PM, X'N: XN,.S'G C SG:S' - S::X'N +XN: X' or2CC: SS':: 2CN: XX. Alternately, 2CGC 2!N S5': XX; or CC: CN:: CS: CX, N.) NT - - XN; JI. Cor.).o. CN - CN or 2VG O N: ON:: CS': CA'. (Prop. I:: CS' -CA': CA'; BC': AC2. PROP. X. If RN be the ordinate of any point P on the hyperbola; then RN'2: AN. A'N: BC': A C2. For NC: NC:: BC': AC'. And rectangles of the same altitude are to one another as their bases, (Euclid, VI. 1.)..TN. N: TN.NC:: BC': AC'. or RN'2: TN. N C:: BC2: A C2. But TN. CN = ON2 - CT. ON, (Euclid, II. 2) = CN - CA', (Prop. VIIIL) = AN. A'N, (Euclid, II. 6.)..N2:AN. A'N::BC': A C2. PROP. XI. If the normal at any point P of an hyperbola meet the transverse axis in G; then,SG: SR:: CS: CA. G 82 CONIC SECTIONS. Produce S'P to W; then since PG bisects the angle SP W, (Prop. IX.) SG: 'G:: SP: S'P,.. SG: S'G-SG:: SP: S'P-SP, but S'P- SP= AA', (Prop. III.) and S'G - SG = SS',.. SG: S'::SP: AA', or SG: SP:: SS': AA', or SG: SP:: CS: CA. CoR. Hence also, S'G: S'P:: CS: CA. PrOP. XII. 53. If from the foci S and S' of an hyperbola SY and S' Y' are drawn at right angles to the tangent at P, then 'Y and Y' are on the circumference of the circle described on AA' as diameter, and SY. S'Y' = BC. Join SP, S'P, and produce SY to meet S'P in W; join CY; then since the angle SPY = the angle WP Y, (Prop. VI.) and the angle S YP = the angle WYP, and the side P Y is common to the triangles SPY, WP Y,.. the triangle SPY = WP Y in all respects,.'. SP= PW, and SY = WY,.S. S'P- SP= S' but S'P- SP = A A', (Prop. III.). '. 8' = AA'. Again,.. SC= CS', and SY= WY,.'. SC: CS':: SY: YW,.C. CY is parallel to S' W,.. CY: SW:: CS:: SS', CONIC SECTIONS. 83 D1) ~\\.CY=- S'W= CA; so CY= CA.. Y and Y' are points on the circumference of the circle described upon A A' as diameter. Next, let S Y be produced to meet this circle in Z, and join ZY'; then since the angle ZYY' is a right angle.. Z Y' passes through the centre C,.the angle SCZ= the angle S'C Y', SZ = S'Y',. SY. S'Y' SY.SZ, = SA. SA', (Euclid, III. 36 Cor.) CS2 -CA2, (Euclid, II. 6.) - B C. COR. If CD be drawn parallel to the tangent at P meeting S'P in E; then since CEPY is a parallelogram,.. PE= CY= AC. G2 84 CONIC SECTIONS. PROP. XIII. 54. To draw a pair of tangents to an hyperbola from an external point 0. Of the foci S and S', let S' be that which is nearer to 0. With centre S and radius equal to AA' describe a circle. Join OS, OS'; and let S O or S 0 produced meet the circle in the point 1. Now if 0 be a point inside the circle MIM' it is evident that OS' is greater than 0I; and if 0 be outside the circle, since OS - OS' < AA' or SI, (Prop. IV.).. OS- OS' < OS-0I,.. OS'> OI With centre 0 and radius OS' describe another circle cutting the former in the points M and M', which it will always do since OS' is greater than 0I. CONIC SECTIONS. 85 Join SMi, SM', and produce them to meet the hyperbola in the points P and P'. Join OP, OP'; these will be the tangents required. Join S'P, S'P'; then since S'P- SP= AA' = SM,. '. S'P=PM. And.. S'P, PO = MP, PO, each to each, and OS' = OM,. the angle OPS' = the angle OPM,.. OP is the tangent at P. (Prop. VI.) So OP' is the tangent at P'. The points of contact P and P' will be upon the same or opposite branches of the hyperbola according as SM and SM' require to be produced in the same or in opposite directions with respect to S, in order to intersect the hyperbola. PROP. XIV. If from a point O a pair of tangents, OP, OP' be drawn to an hyperbola, then the angles which OP and OP' subtend at either focus will be equal or supplementary according as the points of contact are in the same or opposite branches of the hyperbola. Let the points P and P' be on opposite branches of the hyperbola. Join PS, S'P; SP', S'P'. Produce PS to M, making PM equal to PS'. Also from P'S cut off a part P'M' equal to P'S'. Join OMi, OM'; S, OS'. Then since OP, PS' = OP, PM, each to each, and the angle OPS' = the angle OPM, (Prop. VI.).. S' = 01[, and the angle O S'P = the angle 0 MP. 86 CONIC SECTIONS. So OS,' OM' and the angle OS'P' = the angle OM'P', OK= QMVI'. Again, -. - SM = S'P - SP Air, and SM' = SP' - S7' = AA', Sm= sM'. And. OS, SM = OS, SM', each to each, and OM = 03', the angle 0 SM = the angle 0 SM', and the angle 01M = the angle OM'S. But O SM is the supplement of 0 SP, and 011f' S is the supplement of O M'P', *,. O SM' is the supplement of 0 SP, and OMP the supplement of OM'P'. But O MP = OS'P and OM'P'= OS'P', *. 0S'P is the supplement of OS' P CONIC SECTIONS. 87 Hence the angles which OP and OP' subtend either at S or B' are supplementary. In a similar manner if P and P' are on the same branch of the hyperbola, the angles subtended either at S or 5' may be shown to be equal. PROP. XV. 55. If the, tangent at any point P of an hyperbola meet the conjugate axis in the point t, and Pn be drawn at right angles to GB; then On. Gt = BC2. I, -raw PNat right angles to GA; then Ct: GT:: PN: NT,... Ct: PN:: CT: NT, '. CGt. Cn: PN2:: CT. CGN: GN. N T; or Ct.Gn Cn: GT. CGN:: PN2: GN.NT,:: BG2: A C2. (Prop. X.) But GT. CN = A C2, GCt. Cn = B C2. 88 CONIC SECTIONS. 56. The proofs that we have given up to this point of the properties of the hyperbola are closely analogous to the corresponding propositions in the ellipse. The remaining properties of the hyperbola are more conveniently investigated by means of its relation to certain lines, which we shall presently define, called Asymptotes, in the same manner as many of the properties of the ellipse were deduced from those of the auxiliary circle. DEF. The hyperbola described (see fig. Prop. XIV.) with C as centre, and BB' as transverse axis, and AA' as conjugate axis, is called the Conjugate Hyperbola. Its foci, which will be on the line BCB', will evidently be at the same distance from C as those of the original hyperbola, since CS2= CA2 4. C.B2 PnoP. XVI. If through any point R on either of the diagonals of the rectangle formed by drawing tangents to the hyperbola and its conjugate at the vertices, A, A', B, B,' two ordinates RPN, R D M, be drawn at right angles to AA' and BB', and meeting either the hyperbola or its conjugate in the points P and D; then NT2 PN2 = -B C and RM2 DM2 = A C2. Let R be a point on the diagonal O'CO; then RN2: CN2:: AO02 AC2,:: BC: AC2, and PN: CN -CA2:: BC2: A C2;.. N2-PN2: CA2:: BC2: AC2;.. 2 - PN = BC2. Again, RM2: CM2:: AC2: BC2, and DM:CM2- CB:: AC2: BC2;.. M2 -DM': BC2:: AC: BC2; R. _ )M-DM= A C'. (Prop. X.) (Prop. X.) CONIC SECTIONS. 89 In exactly the same manner, if' NB had been produced to meet the conjugate hyperbola in P, and 1MR had been produced to meet the original hyperbola in D, we should have had, PN12 RNy = BC2, and DM2 RM2= A C2. CoR. If RP be produced to meet the hyperbola in p, and the other asymptote in r; then RN2 - PN2 = BP. Pr: (Euclid, II. 5) P. RP. r= BC2. Hence as RPN is further removed fiomn A, and the line Pr consequently increases, since the rectangle contained by RP and Pr remains constant, RP must diminish, and by taking R sufficiently far from C, RP may be made less than any assignable magnitude. The line CR, therefore, continually approaches nearer and nearer to the hyperbola, though it never actually reaches it. 90 CONIC SECTIONS. On account of this property, CR is called an Asymptote to the hyperbola. So also if P be the point where NR produced meets the conjugate hyperbola, we shall have RP. Pr = BC; and therefore CR is also the asymptote to the conjugate hyperbola. In the same manner it may be shown that the other diameter o Co' of the rectangle 0 0' is an asymptote to both hyperbolas. PROP. XVII. 57. If E be the point where the asymptote meets the directrix; then CE= A C CONIC SECTIONS. 9 91 For by similar triangles, CE: 00:: CX: CA, CA: CS. (Prop. 1I.) But 6C02 CA' ~ CB2 = CS'; CO= CS; CE= AC. Com. If SE be joined, since 0112 CA2 CS. CX, the angle 0115 is a right angle. (Eaclid, VI. 8, Cor.) PROP. XVIII. If from any point B in one of the asymptotes to an hyperbola ordinates BPN, BDM be drawn to the hyperbola and its conjugate respectively, and PD be joined, PD will be parallel to the other asymptote. For RN2: BM2:: BC2 AC2; and RN2 -PN2: Bl2' - D1i2:: BC': A C, (Prop. XVI)..'.PXJ D:DM2::BCO:'AC4;::RI2: p31~2,..PN: DMN::B:BM; *. PD is parallel to iMiXi (Euclid, VI. 2.) Also CIVO: CM:: AC:BC, i,. MN is parallel to AB; and OA: Ao:: OB: Bo', A,. BA is parallel to oo'. Hence PD is parallel to oo'. COR. So also if R and D be the points where NIT and 21111 produced meet respectively the conjugate and the original hyperbola, PD will be still parallel to 00o. 92 CONIC, SECTIONS. PROP. XIX. 158. if through any two points Q and Q' of an hyperbola a line 1?RQQ'BR' be drawn in any direction meeting the asymptotes in NR and B'; then will BRQ'= B'Q. I 7' 7 UR U7 L 3// 171 pi ~ ' I/ wl, Throngh Q and Q' draw the ordinates UQ W, U'Q'W; meeting the asymptotes in U, W, U', W'; then by similar triangles, QR: QU:: Q'B: Q'U, and QB' QW:: Q'B' Q: ' *compounding QBR. QR': Q U. Q W:: Q'R. Q'J( Q'U'. Q'W'. But Q U. QW= B C2 = Q'U'. Q'W (Prop. XVI. C'or.) Q. QB = Q B. Q'B,; CONIC SECTIONS. 93 but QR. QR' =QR. QQ' QR. Q'Re' and Q'R. Q'B' = Q'R'. QQ' - QR. Q'R';..Q. Q QQ' Q''. QQ'. Q. Q = Q'B'. COR. 1. If R Q Q'R' move parallel to itself until the points Q and Q' coincide, the line RQ ' will ultimately assume the position LP l, and will become a tangent to the hyperbola at P. Hence, since B Q is always equal to B' Q', LP = P, or the tangent LPl is bisected at the point of contact P. COR. 2. If CP be produced to meet RR' in V, then since RV: VE':: LP: PI, RV== VIR'; and R Q = Q'R',.. QV= Q'V. Hence if a series of parallel chords be drawn in an hyperbola, their middle points will all be in the line drawn through the centre and the point where the tangent parallel to the chords meets the hyperbola. DEF. A line PCP' drawn through the centre, and meeting the hyperbola in P and P', is called a Diameter. A diameter consequently bisects all chords drawn parallel to the tangents at its extremities. PROP. XX. 59. If through any point Q of an hyperbola a line B Qr, be drawn in any direction meeting the asymptotes in R and r, and LPI be the tangent drawn parallel to BQr; then Q. Qr = PL. Through P and Q draw the ordinates EPe, UQW, meeting the asymptote in E, e, U, W; then by similar triangles, 94 CONIC SECTIONS. QB: QU:: PL: PE, Qr:QWI:: P1: Pe QPT. Qr: QU. QW:: PL. PI: PE. Pe but QU. Ql = B C2 PE. Pe. (Prop. XVI. Cor.) QR. Qr=PL. PI, PL'. (Prop. XIX. Cor. 1.) CoR. If Qq be produced to meet the conjugate hyperbola in Q', q' we may show that Q'R. Q'r - PL2, and also, as in Proposition XIX., that Q'IR = qr, Q Q&& = qq.~ Hence if a line be drawn in any direction meeting both the hyperbolas, the portions intercepted between the hyperbola and its conjugate will be equal. CONIC SECTIONS. 901 - PROP. XXI. 60. If from any point P of an hyperbola, PHf and PK be drawn parallel to the asymptotes, meeting them in H and K respectively; then 4. PH: PK = CS2. L 0 r1 Draw the ordinate 1? PN-r meeting the asymptotes in 1? and r; then by similar triangles, PH: PR C: o Oo, and PK: Pr:: CO: Oo, PH. PK:PR. Pr::C02:o, CS2: 4BC2. But PB. Pr B= BC2, 4 PH. PK= CS'. 96 CONIC SECTIONS. PROP. XXII If the tangent at any point P of an hyperbola meet the asymptotes in L and I; then the area of the triangle L C1 is equal to the rectangle contained by A C and B C. Draw PH and PK parallel to the asymptotes meeting them in H and K; then since CL: CH:: LI: Pl, and LI = 2 PI, (Prop. XIX. Cor. 1.).CL = 2 CH= 2 PK; so Cl= 2 CK- 2 PH,.CL. Cl = 49PH. PK = CS2 (Prop. XXI.) = CO. Co,.CL CO:: Co: CI,. the triangles LCI, OCo have the angle at C common and the sides about those angles reciprocally proportional;. the triangles L C1 = the triangle O Co, = A C. A 0. = AC. BC. PROP. XXIII. 61. If from any point R in the asymptote of an hyperbola, two ordinates RPPN and DRDM be drawn to the hyperbola and its conjugate respectively, then the tangents at P and D will be parallel respectively to CD and CP. Join PD, meeting CR in H; then since PD is parallel to oo', (Pop. XVIII.) the tangents at P and D will each meet CR produced in the same point L. (Prop. XXII.) Produce LP and LD to meet the other asymptotes in I and 1'; then since CL. Cl = CS' = CL. Cl', (Prop. XXII.) CONIC SECTIONS. 97.. Cl= C',... IC: Cl':: P: PL,.'. CP is parallel to the tangent at D. Also I'D: DL:: TC: C1,.~. CD is parallel to the tangent at P. The lines CP and CD are called Conjugate Diameters, since each of these lines is parallel to the tangent at the extremity of the other. PROP. XXIV. If CP and CD be semi-conjugate diameters in the hyperbola; then CP', CD2 = CA't - CB2 Draw the ordinates NPiR, MDR meeting the asymptote in the point R (Prop. XXIII.); then H 98 CONIC SECTIONS. CR2- C2= NBR2 -NP2, = BC2, (Prop. XVI.) CB2 = CP2~BC2; so CR2 CD2 2 AC2, CP2+ BC2 CD2= 2AC; or CP2 CD2 =4 C' B BC2. PROP. XXV. (2. The area of any parallelogram formed by drawing tangents to the hyperbola and its conjngate at the extremities P, P', D, D' of' a pair of conjugate diameters is equal to the, rectangle contained by the axes. i 7) I) Let L iL'1' be the parallelogram formed by drawing tangents at the extremities P, P', D, DR of any pair of conjugate diameters. The points L, L', 1, ', will (Prop. XXIII.) be on the asymptotes. CONIC SECTIONS. 9 99 Now the parallelogram L-L' 4 parallelogram CL, 4 triangle L Cl1, 4 A C. B C, (Prop. XXII.) =AA'. BB'. CoR. If PE be drawn perpendicular to CD, then PF. CD =A C. BC Also, if -the normal PG meet the transverse axis in U, ag in the ellipse, P.G BC' 63. DEF. The line Q V drawn from any point Q of the hyperbola parallel to the tangent ait any point P, and meeting CP produced in 17; is called an Ordlinate to the diameter CP. Pu'op. XXVI. If Q V be, an ordlinate to the diameter P'CP, and CD be conjugate to CP; then Q V2 P1V. PTV:: CD'2: CP2. Produce VQ to meet the, asymptotes in B and r; and let the tangent at P meet the asymptotes in L and 1; then By': PD:: C V': CP', RBV' — PL':PL:C CV' - CP': CP'. But BRQ. Qr- PL', (Prop. XX.) VV-T' zPL', or RV17' PL'2 Q IT AdCV' _ CP' P17. P'T7 (Euclid, II. (3.) QV': PL':: P17. P'V: CP2. Alternately, Q1V': P V. P'V:: PL':C. But since PD is a parallelogram, (Prop. XXIII.) PL = CD. Hence QT72: PV. P'V:: CD'2: CP'. CoR. If TVQ be produced to meet the conjugate hyperbola in QV, then since Q'B. Q'r = PL', (Prop. XX. (Cor.) Q' V2 _BV '1 _PL2. Hence Q' V2: CPV'+ OP':: CD2: Cpl. H 2 100 CONIC SECTIONS. PiOP. XX VII 64. If QV be an ordinate to the diameter PV, and the tangent at Q meet CP in the point T; then CV. CT= CP. Draw the tangent LPI meeting the asymptotes in the points L, 1; also let the tangent at Q meet the asymptotes in X, r. Draw R K, r k parallel to Q V meeting CP in K, k. Now since the triangles JRCr, L Cl are equal, (Prop. XXII.) and have the angle at C common,.. CR: CL:: C1: Cr. (Euclid, VI. 15.) But CR: CL:: CK CP, CONIC SECTIONS. 101 and C: Cr:: CP: Clt,. CK: CP::CP: C, CK.. Ck= CP. Again, produce BK and Q V to meet the asymptote C1 in R' and q; then Since Rr is bisected in Q, (Prop. XIX. Cor. 1).. 'r is bisected in q, and R- K I 'K, (Prop. XIX. Cor. 2).'. K is parallel to Br,.. T: C:: Cr: Cq,:: C: CV,.CV. CT= CK. Ck CP2. CoR. 1. Conversely, if Q V be an ordinate to PV, and CV. CT= CP2, then Q T is the tangent at Q. COR. 2. Hence also, if RR' meet the curve in U and U', and k U, kU' be drawn, since CK. Ck = CP2, k U and k U' are tangents to the hyperbola at U and U'. PnOP. XXVIII. 65. If two chords of a hyperbola intersect one another, the rectangles contained by their segments are proportional to the squares of the diameters parallel to them. Let QOq be any chord drawn through the point 0, and let CD be drawn parallel to it, meeting the conjugate hyperbola in D. Produce Qg to meet the asymptotes in R and r; and draw the diameter CPV, bisecting both Qq and Rr in V. (Prop. XIX. Cor. 2.) Also draw the tangent L P parallel to Qq, meeting the asymptotes in L and 1. 102 102 ~~~CONIC SECTIONS. Now since Qq is divided equally in V and unequally in 0,. QO0. O q = Q V72 _ OV; (Euclid, II. 5) so also P0. Or =B1 V2 - 0T72, (Euclid, TI. 5) B.O. Or-QO. Oq=1RV-Q B Q. Q r (Euclid, IT. 5) PL2, (prop. XX.) Q0. Oq=PO. Or -PL'. Again, through 0 and P draw E 0 e, UP TY~ at right angles to the axis meeting the asymptotes in E, e, U, W; then B0 OE:: PL: PU, andrO O 0e PI1 P W, P 0. rO0 OE. Oe PL2: P U. P W; CONIC SECTIONS. 103 but PU. PW = B C, (Prop. XVI.) and PL2 = CD2, (Prop. XXIIL). O.rO: OE. Oe:: CD2: BC, or O.rO: CD2:: OE. Oe: BC2,. RO.rO-PL2: CD2::OE. Oe -BC2: BC, or QO. Of: CD2:: OE. Oe -BC2: BC2. In the same manner if through 0 another chord Q' Oq' be drawn, and CD' be drawn parallel to it, meeting the conjugate hyperbola in D', we shall have Q'O. Oq': CD'2:: OE. Oe-BC2: BC2. Hence QO. Oq: Q'O. Oq':: CD: CD'2. Con. The same result may be shown to be true when the point 0 is outside the hyperbola. Moreover, it is not necessary that the chords should be drawn meeting one branch only of the hyperbola or the same branch. The proportion still holds good when one or both of the chords meet both branches of the hyperbola, or when the chords are drawn in different branches. 66. DEF. If with a point 0 on the normal at P as centre. and OP as radius, a circle be described touching the hyperbola at P, and cutting it in Q; then when the point Q is made to approach indefinitely near to P, the circle is called the Circle of Curvature at the point P. PROP. XXIX. If PH be the chord of the circle of curvature at the point P of a hyperbola, which passes through the centre; then PH. CP = 2 CD2. Let PT be the tangent, and PG the normal at the point P. With centre 0, and radius OP, describe a circle cutting the hyperbola in the point Q. Draw B Q W parallel to CP, meeting the circle in W, and TP produced in R. Also, draw Q V parallel to PR, meeting the diameter PP' in V; then since RP touches the circle at P, 104 CONIC SECTIONS. BQ. RQ. R W= PR2, (Euclid, 1II. 36) or PV. RW= Q V2. But Q 1V: P17. P 17:: CD2: UP2, (Pr1op. XXVI.) PV. RW: PV. P'V:: CD2: 2 P2, or R i: PFV:: CD': C]?. Now, when the circle becomes the circle of curvature at P, the points B and Q move up to, and coincide with P, and the lines RIW and PH become equal, while P' V becomes equal to PP', or 2 UP. Hence, PH: 2 P:: CD2: UP1,..PH. CP: 2CP2::2CD:2CP2,.PH. CP =2 CD2. PRoP. XXX. If P U be the diameter of the circle of curvature at the point P of the hyperbola, and PP he drawn at right. angles to CD; then PU. P -= 2CD2 CONIC SECTIONS. 105 Since the triangle PHU is similar to the triangle PFC,.PU PH:: CP: PF,. PU. PF = PH: CP, = 2 CD. (Prop. XXIX.) PorP. XXXI. If PI be the chord of the circle of curvature through the focus of the hyperbola; then PI. AC= 2CD'. Let S'P meet CD in E; then since the triangles PIU and PEF are similar,. PI: PU:: PF: PE. But PE A C, (Prop. XII. Cor.). PI: PU:: PF: AC,. PI. AC = PU. PF, =2 CD. (Prop. XXX.) The point where the circle of curvature intersects the hyperbola may be determined as in the case of the ellipse. PnOP. XXXII. 67. If P be any point on the hyperbola, and CD be conjugate to CP; then SP. S'P CD2. Draw PIT' parallel to the asymptote CE meeting the directrices in I and P', and CB' in U. Let the ordinates, NP, MD meet the asymptote in R, and draw PW perpendicular to the directrix; then by similar riangles, PI: PW: CE: CX, CA: CX. (Prop. XVII.) 106 CONIC SECTIONS. But P: PW:: SA: AX,:: CA: CX... SP = PI; so S'P = PI', SP.. S'P = PI. PI', = UP - U2, = CR2 - CE2, = CR" - CA. (Prop. XVII.) But CR - CD" = R -f2 DDM2, = CA2, (Prop. XVI.).C. CR2- CA= CD2. Hence SP. S'P = CD. PROBLEMS ON THE HYPERBOLA. 1. THE locus of the centre of a circle touching two given circles is an hyperbola or ellipse. 2. If on the portion of any tangent intercepted between the tangents at the vertices a circle be described, it will pass through the foci. 3. In an hyperbola the tangents at the vertices will meet the asymptotes in the circumference of the circle described on SS' as diameter. 4. If from a point Pin an hyperbola P11' be drawn parallel to the transverse axis meeting the asymptotes in I and I'; then PI. P' = A C2. 5. If a circle be inscribed in the triangle SPS', the locus of its centre is the tangent at the vertex. 6. If PN be the ordinate of the point P, and NQ a tangent to the circle described on the transverse axis as diameter, and PM be drawn parallel to Q C meeting the axis in M, then MNT = B C. 7. If PN be the ordinate of a point P, and NQ be drawn parallel to AP to meet CP in Q, then A Q is parallel to the tangent at P. 8. If an hyperbola and an ellipse have the same foci, they cut one another at right angles. 9. If the tangent at P intersect the tangents at the vertices in R, r, and the tangent at P' intersect them in R', r', then AS. Ar = AR'. Ar'. 108 CONIC SECTIONS. 10. If any two tangents be drawn to an hyperbola, the lines joining the points where they intersect the asymptotes will be parallel. 11. The perpendicular drawn from the focus to the asymptotes of an hyperbola is equal to the semi-conjugate axis. 12. If the asymptotes meet the tangent at the vertex in 0, and the directrix in E; then AE is parallel to S 0. 13. In a rectangular hyperbola conjugate diameters are equal to one another. 14. In a rectangular hyperbola the normal PG is equal to CP. 15. The lines drawn from any point in a rectangular hyperbola to the extremities of a diameter make equal angles with the asymptotes. 16. Prove that the asymptotes to an hyperbola bisect the lines joining the extremities of conjugate diameters. 17. A line drawn through one of the vertices of an hyperbola and terminated by two lines drawn through the other vertex parallel to the asymptotes will be bisected at the other point where it cuts the hyperbola. 18. P is any point on an hyperbola, and P' a point on the conjugate hyperbola. If CP and CP' be conjugate, prove that S'P - SP = A C- B C, S and S' being the interior foci. 19. If CP and CD be conjugate, and through C a line be drawn parallel to either focal distance of P, the perpendicular from D upon this line is equal to B C. 20. Given a pair of conjugate diameters, find the principal axes. 21. If Q be a point on the conjugate axis of a rectangular hyperbola, and QP be drawn parallel to the transverse axis meeting the curve in P; then PQ = A Q. 22. In a rectangular hyperbola the focal chords drawn parallel to conjugate diameters are equal. CONIC SECTIONS. 109 23. If in an equilateral hyperbola CY be drawn at right angles to the tangent at P, and A Y be joined, the triangles PCA, CA Y are similar. 24. The radius of the circle which touches an hyperbola and its asymptotes, is equal to that part of the latus rectum produced which is intercepted between the curve and the asymptotes. 25. If Q Q' by any chord of an hyperbola, and CP the diameter corresponding to it, and Qf, PK, Q'H' be drawn parallel to one asymptote meeting the other in H, K and H', then CH. CH' = CK2. 26. If the chord RPP'R' intersect the hyperbola in the points P, P', and the asymptotes in B, R'; and PK be drawn parallel to OC', and P'K' to CR; then IK = P'K', and 'K' = PK. 27. If AA' be any diameter of a circle, and PNQ an ordinate to it, then the locus of the intersections of AP, A'Q is a rectangular hyperbola. 28. If two concentric rectangular hyperbolas be described, the axes of one being the asymptotes of the other, they will intersect at right angles. 29. If any chord AP through the vertex be divided in Q, so that A Q: QP:: AC': BC, and QN be drawn to the foot of the ordinate PV, prove that a straight line drawn at right angles to QN from Q cuts the transverse axis in the same ratio. 30. Prove that the curve which trisects the arcs of all segments of a circle described upon a given base is an hyperbola. 31. If SVs, TVt be two tangents cutting one asymptote in S, T, and the other in s, t, prove that VS: Vs: Vt VT. 32. If from the exterior focus of an hyperbola a circle be described with radius equal to BC, and tangents be drawn to it from any point in the hyperbola, the line joining the points of contact will touch the circle described on the transverse axis as diameter. 110 CONIC SECTIONS. 33. Circles are drawn touching the straight line A B in a fixed point C; and from the fixed points A, B tangents are drawn to these circles. The locus of their intersection is an ellipse or hyperbola. Distinguish between the two cases. 34. PP' is a double ordinate in an ellipse. A P, A'P' are produced to meet in Q. Prove that the locus of Q is an hyperbola with the same axes as the ellipse, 35. If the tangent at P intersect the asymptotes in L and 1, and PG be the normal at P, then the angle LG1 is a right angle. 36. If an ellipse, a parabola, and a hyperbola, have an common tangent, and the same curvature at the vertex, the ellipse will be entirely within the parabola, and the parabola entirely within the hyperbola. 37. The chord RPP'R' of an hyperbola intersects the asymptotes in R and R'. From the point R a tangent RQ is drawn meeting the hyperbola in Q. If PH, QK, P'H' be drawn parallel to one asymptote meeting the other in the points H, K, Ht'; then PH + P'H' = 2 QI. 38. If through P, P' on an hyperbola lines be drawn parallel to the asymptotes forming a parallelogram, of which PP is one diagonal; the other diagonal will pass through the centre. 39. If P be the middle point of a line EF which moves so as to cut off a constant area from the corner of a rectangle, its locus is an equilateral hyperbola. 40. Pll, PN are drawn parallel to the asymptotes CN, Cill, and an ellipse is constructed having CX, C1f for semiconjugate diameters. If CP cut the ellipse in Q, the tangents at Q and P to the ellipse and hyperbola are parallel. 41. If a circle be described through any point P of a given hyperbola and the extremities A, A' of the transverse axis, and IVP be produced to meet the circle in Q; prove that Q traces out an hyperbola whose conjugate axis is a third proportional to the conjugate and transverse axes of the original hyperbola. CONIC SECTIONS. 111 42. If lines be drawn from any point of a rectangular hyperbola to the extremities of a diameter, the difference between the angles which they make with the diameter will be equal to the angle which this diameter makes with its conjugate. 43. If between a rectangular hyperbola and its asymptotes any number of concentric elliptic quadrants be inscribed, the rectangle contained by their axes will be constant. 44. In the rectangular hyperbola if CP be produced to Q so that PQ = CP, and Q 0 be drawn at right angles to Q to intersect the normal in 0, 0 is the centre of curvature at P. 45. With two conjugate diameters of an ellipse as asymptotes a pair of conjugate hyperbolas are constructed; prove that if one hyperbola touch the ellipse the other will do so likewise; prove also that the diameters drawn through the points of contact are conjugate to each other. 46. If a pair of conjugate diameters of an ellipse when produced be asymptotes to an hyperbola, the points of the hyperbola at which a tangent to the hyperbola will also 'be a tangent to the ellipse, lie in an ellipse similar to the given one. 47. In the rectangular hyperbola the radius of curvature at P is to the radius of curvature at P' in the triplicate ratio of CP to CP'. 48. OP, OQ are tangents to an ellipse at P and Q, and asymptotes to an hyperbola. Show that a pair of their common chords is parallel to PQ. One of these chords being RS, prove that if PR touches the hyperbola at P, Q S touches it at S; also if PS, QR meet in U, OU bisects PQ. 49. The base of the triangle ABC remains fixed, while the vertex C moves in an equilateral hyperbola passing through A and B. If P, Q be the points in which AC, BC meet the circle described on AB as diameter, the intersection of A Q, BP is on the other branch of the hyperbola. CHAPTER IV. THE SECTIONS OF THE CONE. 68. DEF. If two indefinite straight lines 10.[', DOD', intersect one another at a point 0, and one of them IOI' remain fixed while the other DOD' revolves round it in such a manner that its inclination to IO' is the same in all positions, the surface generated by D OD' will be a Right Cone. The line 1IO' is called the Axis, and the point 0 the VIertex of the Cone. It now remains for us to show (see Introduction) that the curve formed by the intersection of this surface with a plane is in general one of the three curves whose properties we have been investigating, and to consider under what circumstances it will be the Parabola, Ellipse, or Hyperbola. If the cutting plane pass through the vertex of the cone as KOK', and intersect the cone again at all, it will in general cut it in two straight lines as OK, OK' which will represent two positions of the generating line. The inclination of these lines to each other will depend upon the inclination of the cutting plane to the axis of the cone, and will be greatest when this plane passes through the axis, in which case it will be double the constant angle between the axis and the generating line. If the cutting plane pass through a generating line dod' and be perpendicular to the plane containing this line and the axis, it will simply touch the cone along this line. CONIC SECTIONS. 113 d D Should the cutting plane not pass through the vertex, and be at right angles to the axis of the cone, the section will evidently be a circle. In any other case the section will, as we proceed to show, be a Parabola, Ellipse, or Hyperbola. Whatever be the position of the cutting plane with respect to the cone, we can always suppose a plane drawn through the axis of the cone at right angles to it; and it will be convenient to have this latter plane represented by the plane of the paper as D Od. The cutting plane will therefore always be taken at right angles to the plane DOd of the parer. I 114 CONIC SECTIONS. PROP. I. 69. The curve formed by the intersection of the surface of a right cone with a plane (which neither passes through its vertex nor is at right angles to its axis) will be a Parabola, Ellipse, or Hyperbola, according as the inclination of the cutting plane to the axis of the cone is equal to, greater, or less than the constant angle which the generating line forms with the axis. Let the plane of the paper represent the plane drawn through the axis IOI' of the cone at right angles to the cutting plane; and let it intersect the surface of the cone in the two generating lines OD, Od. Let the cutting plane intersect the surface of the cone in the curve PA, and the plane of the paper in the line ANH. The curve will evidently be symmetrical with respect to this line. On AH take any point N, and through N draw a plane perpendicular to the axis meeting the surface of the cone in the circle RPr, and the cutting plane in the line PN, which will be at right angles to the plane of the paper and to AN. Let a sphere be inscribed in the cone touching the cone in the circle EQe and the cutting plane in the point S, and let the plane EQe intersect the cutting plane in the line XM, which will be at right angles to the plane of the paper, and therefore parallel to PN. Draw PM perpendicular to XM, and join >PS, PO, and let PO meet the circle EQe in the point Q.* Then since PS and P Q are both tangents to the sphere,.'. PS=PQ. But PQ = RE,. PS= RE. * N.B. In the figure, to avoid confusion, that part of the section which lies above the plane of the paper is alone represented. CONIC SECTIONS. 1 I 0 / \ // a\R // /Xi'/ I ' But RE: XN:: A:E AX, (Euclid, VI. 2.) and AE- AS,.'. RE: XN:: AS: AX,.. SP:P,:: AS: AX,.'. the curve PA is either a Parabola, Ellipse, or Hyperbola, whose focus is S and directrix XM. Again, let AH meet the axis 01 in F. Then the angle AFO will be the inclination of the cutting plane to the axis. (1) Let the angle AFO = the angle FOd; then AHI is parallel to Od,.. the angle AXE = the angle OeE = the angle AEX,.. AE=AX,.. AS=AX,.'. the curve AP will be a Parabola. I 2 116 CONIC SECTIONS. A (:i A' X' E' CONIC SECTIONS. 117 (2) Let the angle AFO be > FOd; then the complement FXE is < the complement OEe,. the angle AXE is < AEX,. AE is < AX, or AS is < AX,. the curve AP is an Ellipse. Since the angles HFO, FOd are together less than tIFO. OFA, i.e. than two right angles, the lines AH and Oe may be produced to meet in A'. If another sphere be described touching the cone in the circle E'Q'e' and the cutting plane in the point S'; and the line X'M' denote the intersection of the plane E'Q'e' with the cutting plane, and PM' be drawn at right angles to this line, it can easily be shown that S'P: PM':: 'A': A'X'. Hence S' and X'M' represent respectively the other focus and directrix of the ellipse. Also if B C be the semi-axis minor, and through the centre C a line UCU' be drawn parallel to Ee meeting OD, Od in U and U', then it is evident that BC2 = CU. CU'. (3) Let the angle AFO be < FOd; then the angle AXE is > the angle AEX,. AE is > AX,.AS is> AX,.. the curve PA is an Hyperbola. Since the angles AFO, FOd' are less than the two FOd, FOd', i.e. than two right angles, the lines FA and dO may be produced to meet in A'. In this case the cutting plane will intersect the other half of the cone, and if any point P' be taken on this part of the curve, and P'M be drawn at right angles to XM, it can be shown as before that SP': P'M:: S'A:: AX. i18 CONIC SECTIONS; d d K CONIC SECTIONS. 119 The intersection of the cutting plane therefore with this portion of the cone will be the other branch of the hyperbola. Also if another sphere be described touching the upper portion of the cone in E'Q'e', and the cutting plane in S', and the line X'M' denote the intersection of the plane E'Q'e' with the cutting plane, and P'M' be drawn at right angles to this line, it can be easily shown that S'M': P'M':: S'A':A'X'. Hence, S' and X'MI' will represent respectively the other focus and directrix of the hyperbola. COR. 1. In this last case, i.e. when the section is an hyperbola, if a plane OKL be drawn through the vertex of the cone parallel to the cutting plane, meeting the plane of the paper in the straight line OL, and the surface of the cone in the generating line OK; then OL: OK:: OL: OR,:: AN: AR,:: A X: AE, (Euclid, VI. 2):: AX: AS,:: CA: CS, (Chap. III. Prop. II.) where C is the middle point of AA', and therefore the centre of the hyperbola... KOL is half the angle between the asymptotes. (Chap. III. Prop. XVI.) Again, if BC be the conjugate semi-axis, and C U'U be drawn parallel to Rr meeting OD', Od' in U and U', then since CU: AC:: L: OL, and CU': A'C:: rL: OL,.. CU. CU': AC2:: r L.rL: OL',:: KL: OL2; butBC2: AC2:: KL: OL2,. BC2 = CU. CU'. CoR. 2. If the cutting plane is parallel to the axis OL and 01 coincide. 120 CONIC SECTIONS. In this case half the angle between the asymptotes of the hyperbolic section is equal to the constant angle D I, and we can at once see that 0 C is the semi-conjugate axis. This affords a convenient method of obtaining a pair of conjugate hyperbolas. Draw Oi at right angles to 01 in the plane of the paper, and let another cone be formed by supposing OD to revolve round Oi in such a manner that the angle DOi is the same in all positions, and equal to the complement of DO I. Then if through any point A on the common generating line OD we draw two planes at right angles to the plane of the paper, and parallel respectively to OI and Oi, they will cut the cones in two hyperbolas, whose semi-transverse axes will be respectively A C, 0C, and whose semi-conjugate axes will be respectively 0C, AC, and which therefore will be conjugate to each other. CONIC SECTIONS. 121 70. As long as the cutting plane remains parallel to itself, it is evident that the ratio of AE to AX, and therefore of AS to AX will be altered. Hence the sections made by planes inclined at the same angle to the axis of the cone will have the same eccentricity.* 71. Through any point Q on the circle EQe let a plane be drawn parallel to ANP, intersecting the plane of the paper in the straight line WLN', the cone in the curve WQP', and the planes of the circular sections EQe, RPr in the ordinates QL, P'N'. Then it is manifest that the curve WQP' will touch the circle wQw', formed by the intersection of the cutting plane with the sphere at the extremities of the ordinate QL produced. Join OP' meeting EQe in Q'; then P"Q': YN'L:: RE:.NX,:: SA: AX. * The ratio of SA: A X, or of CS: CA, is called the eccentricity. 122 CONIC SECTIONS. But P'Q' is equal to the tangent drawn from P' to the circle wQw', and N'L is equal to the perpendicular from P' on the common ordinate of the circle wQw' and the section WQF. Hence we have the following important property,If a circle touch a conic section in two points at the extremities of an ordinate, the ratio which the tangent drawn to the circle from any point on the curve bears to the perpendicular from the same point on the common chord is equal to the eccentricity of the conic section. If the two points in which the circle touches the conic coincide, the circle becomes the circle of curvature at the vertex; and therefore the ratio which the tangent drawn from any point of a conic section to the circle of curvature at the vertex bears to the abscissa of the point, is constant and equal to the eccentricity of the curve. PROBLEMSS ON THE SECTIONS OF THE CONE. 1. THE foci of all parabolic sections which can be cut from a given right cone lie upon the surface of another cone. 2. The foci of all elliptical sections of a given right cone, in which the ratio of CA to CS is the same, will lie on two other cones. 3. The extremities of the minor axes of the elliptical sections of a right cone made by parallel planes lie on two generating lines. 4. The latus rectum of a parabola cut from a given cone varies as the distance between the vertices of the cone and the parabola. 5. Under what conditions is it possible to cut an equilateral hyperbola from a given right cone? 6. Two cones whose vertical angles are supplementary are joined as in Art. 69, Cor. 2. Prove that the latera recta of the curves of section of the cones, whose axes are respectively 01 and Oi, made by planes parallel or perpendicular to the plane of the axes, are in the duplicate ratio of Oi and 1. ADDITIONAL PROBLEMS. 1. SHOW that the part of the directrix of a parabola, intercepted between the perpendiculars on it from the extremities of any focal chord, subtends a right angle at the focus. 2. The locus of the foci of all parabolas touching the three sides of a triangle is a circle. Prove this, and give a geometrical construction for finding the centre. 3. A system of parabolas which always touch two given straight lines have their axes parallel; show that the locus of the foci is a straight line. 4. Prove that the locus at the foot of the perpendicular from the focus of a parabola on the normal is a parabola. 5. If S be the focus of a parabola, which touches the sides A B, A C of the triangle AB C at the points B, C, and 0 the centre of the circle described about the triangle; prove that the angle 0 SA is a right angle. 6. From the focus of a parabola a straight line is drawn parallel to the tangent at any point P, meeting the diameter through P in V'; show that the tangent drawn from P to any circle passing through V' is equal to one-half of the ordinate Q V, V being the second point in which the circle cuts the diameter through P. 7. PSp is a focal chord, and upon PS and p S as diameters, circles are described; prove that the length of either of their common tangents is a mean proportional between A S and Pp. CONIC SECTIONS. 125 8. If A Q be a chord of a parabola through the vertex A, and QR be drawn perpendicular to A Q to meet the axis in B; prove that AR will be equal to the chord through the focus parallel to A Q. 9. The locus of the vertices of all parabolas, which have a common focus and a common tangent, is a circle. 10. Two parabolas have a common axis and vertex, and their concavities turned in opposite directions; the latus rectum of one is eight times that of the other; prove that the portion of a tangent to the former, intercepted between the common tangent and axis, is bisected by the latter. 11. B is a point on a radius OA of a circle, whose centre is 0. On OA produced a point C is taken, such that OB. 00 = OA'. If P be any point on the circumference of this circle, B the middle point of BP, and Q the point of intersection of AR, CP; prove that the locus of Q is a circle. 12. If from the middle point of a focal chord of a parabola two straight lines be drawn, one perpendicular to the chord meeting the axis in G, and the other perpendicular to the axis meeting it in N; show that NG is constant. 13. A circle is drawn touching the axis of a parabola, the focal distance of a point P, and the diameter through P. Show that the locus of its centre is a parabola with vertex S, and latus rectum equal to A S. 14. If from the point of intersection of the directrix and axis of a parabola a chord XPQ be drawn, cutting the parabola in P, Q; show that the rectangle contained by the ordinates of PQ is equal to the square of one-half the latus rectum. 15. Find the locus of the centre of a circle which touches a given circle and a given straight line. 16. Given one point of contact of a parabola with three 126 CONIC SECTIONS. tangents given in position, find the two other points of contact. 17. The triangle ABC circumscribes a parabola whose focus is S. Through A, B, C, lines are drawn perpendicular respectively to SA, SB, SC. Show that these lines pass through one point. 18. From the focus a line is drawn parallel to the tangent at P, meeting the parabola at Q. QN is an ordinate, and the tangents at P and Q meet the axis in T and T'. Prove that SN2 = 4AT. AT', and that if the diameter at P meet SQ in E, the locus of E is a parabola, whose latus rectum is half that of the given parabola. 19. P is any point in a parabola; through S a line is drawn at right angles to the axis, meeting the chord AP or AP produced in R. Prove that SK. SR = 2AS. S Y, where SY is the perpendicular on the tangent, and SK on the normal. 20. From the focus S of a parabola SK is drawn, making a given angle with the tangent at P. Show that the locus of K is that tangent to the parabola which makes with the axis an angle equal to the given angle. 21. PS Q is a focal chord of a parabola, AP' a parallel chord meeting the latus rectum in Q'; prove that AP'. A Q =SP. SQ. 22. The circle of curvature at any point bf a parabola whose abscissa is A N cuts the axis in U and U'. Prove that AU. AU' = 3AN'. 23. AB is a diameter of a circle. From any point Q in the circumference a tangent QP is drawn, and from P a perpendicular PN is let fall upon AB. Show that if P be always taken so that QP is equal to AN, the locus of P will be a parabola. 24. If a tangent be drawn from any point of a parabola to the circle of curvature at the vertex, the length of the tangent CONIC SECTIONS. 127 will be equal to the abscissa of the point measured along the axis. 25. To two parabolas which have a common focus and axis two tangents are drawn at right angles; the locus of their intersection is a straight line parallel to the directrices. 26. If any three tangents be drawn to a parabola, the circle described about the triangle so formed will pass through the focus, and the perpendiculars from the angles on the opposite sides intersect in the directrix. 27. A parabola touches one side of a triangle in its middle point, and the other two sides produced. Prove that the perpendiculars drawn from the angles of the triangle upon any tangent to the parabola are in harmonical progression. 28. Two equal parabolas have the same axis and vertex, but are turned in opposite directions; chords of one parabola are tangents to the other. Show that the locus of the middle point of the chords is a parabola whose latus rectum is onethird of that of the given parabola. 29. Two equal parabolas have the same focus, and their axes are at right angle to each other, and a normal to one of them is perpendicular to a normal of the other; prove that the locus of the intersection of such lines is a parabola. 30. Show that in every ellipse there are two equal conjugate diameters, coinciding in direction with the diagonals of the rectangle, which touches the ellipse at the extremities of the axes. 31. If a circle be described through the two foci of an ellipse, cutting the ellipse, show that the angle between the tangents to this circle, and to the ellipse at either point of intersection, is equal to the inclination of the normal to the ellipse to the axis minor. 32. The points in which the tangents at the extremities of the transverse axis of an ellipse are cut by the tangent at any 128 CONIC SECTIONS. point of the curve, are joined one with each focus; prove that the point of intersection of the joining lines lies in the normal at the point. 33. The external angle between any two tangents to an ellipse is equal to the semi-sum of the angles which the chord joining the points of contact subtends at the foci. 34. The tangent to an ellipse at any point P is cut by any two conjugate diameters in T, t, and the points T, t, are joined with the foci S, S' respectively; prove that the triangles SP; S'Pt are similar to each other. 35. P is any point on a fixed circle, the centre of which is 0; E is a fixed point without the circle; an ellipse is described with centre 0 and area constant so as always to touch EP at P. Find the locus of the extremities of the diameter conjugate to OP. 36. The normal at any point P of an ellipse cuts the axes in G, g; prove that if any circle be described passing through G, g, the tangent to it from P is equal to CD. 37. Given a focus, a tangent, and the eccentricity of a conic section; prove that the locus of the centre is a circle. 38. A straight line is drawn through a given point C within a circle to cut it in P'. If p is taken in it such that Cp2 = OP. CP', find the locus of p. 39. In the ellipse PY. PY': PN2:: C8'2: BC and SY. CD = SP. BC. 40. Show that if the distance between the foci of the ellipse be greater than the length of its axis minor, there will be four positions of the tangent, for which the area of the triangle, included between it and the straight lines drawn from the centre of the curve to the feet of the perpendiculars from the foci on the tangent, will be the greatest possible. CONIC SECTIONS. 129 41. Two conjugate diameters of an ellipse are cut by the tangent at any point P in MU, Nr; prove that the area of the triangle CPM varies inversely as that of the triangle CPN. 42. Circles are described on S Y, S'Y' as diameters, cutting SP, S'P respectively in Q, Q'. Prove that SQ. S'P = SP. S'Q' = BC2. 43. PSP', pSp' are any two focal chords of a conic section, P and p being on the same side of the axis; prove that Pp, P'p' meet on the directrix. 44. Prove that an ellipse can be inscribed in any parallelogram so as to touch the middle points of the four sides; and show that this ellipse is the greatest of all inscribed ellipses. 45. If from any point on the exterior of two concentric, similar, and similarly placed ellipses, two tangents be drawn to the interior ellipse which also intersect the exterior; show that the distance between the points of intersection will be double of that between the points of contact. 46. The tangent at any point P in an ellipse, of which S and H are the foci, meets the axis major in T, and TIQR bisects HP in Q, and meets SP in R; prove that PR is onefourth of the chord of curvature at P through S. 47. Prove that the distance between the two points on the circumference of an ellipse at which a given chord, not passing through the centre, subtends the greatest and least angles, is equal to the diameter which bisects that chord. 48. From any point on the auxiliary circle chords are drawn through the foci of an ellipse, and straight lines join the extremities of the chords with the extremity of the diameter passing through the point; prove that these lines will touch the ellipse. 49. A quadrilateral circumscribes an ellipse. Prove that either pair of opposite sides subtends supplementary angles at either focus. K CONIC SECTIONS. 50. Two tangents to an ellipse intersect at right angles; show that the straight line joining their point of intersection with the point of intersection of the normals at the points of contact passes through the centre. 51. P, Q are points in two confocal ellipses, at which the line joining the common foci. subtends equal angles; prove that the tangents at P, Q are inclined to an angle which is equal to the angle subtended by P Q at either focus. 52. Tangents to an ellipse are drawn from any point in a circle through the foci; prove that the lines bisecting the angle between the tangents all pass through a fixed point. 53. If the ordinate at P meet the auxiliary circle in Q, and CQ meet the ellipse in R, then CR is equal to the perpendicular on the tangent at P from C. 54. If P be a point such that SP, S'P are perpendicular; prove that CD2 = 2. BC2. 55. If circles be described to the triangle SPS' opposite to the angles S and S'; prove that the rectangle contained by their radii is equal to B C2. 56. The circle of curvature at any point P of an ellipse meets the focal distances in, R'; S U is a tangent to the circle. Prove that SU2: SP2:: 2. SP- 3. AC: AC, and if RR' passes through the centre of the circle of curvature, CP = CS. Determine the limits of possibility in both cases. 57. A straight line is drawn from the centre of an ellipse meeting the ellipse in P, the circle on the major axis in Q, and the tangent at the vertex in T. Prove that as CT approaches and ultimately coincides with the semi-major axis, PT and QT are ultimately in the duplicate ratio of the axes. CONIC SECTIONS. 131 58. A straight line is drawn through the focus S of an ellipse meeting the two tangents at right angles to it in Y and Z, the diameter parallel to these tangents in L, and the directrix in M; prove that SL: SY:: SZ: SM. 59. If any equilateral triangle PQR be described in the auxiliary circle of an ellipse, and the ordinates to P, Q, R meet the ellipse in PE', Q', '; the circles of curvature at P', Q', B', meet in one point lying on the ellipse. 60. From a point T two tangents TP, TQ are drawn to an ellipse. Show that a circle with T as centre can be described so as to touch SP, S'P, SQ, S'Q. 61. If the normal at P meet the axis minor in g, and if the tangent at P meet the tangent at the vertex A in V; show that Sg: SC:: PV: VA. 62. If a circle passing through Yand Y' touch the major axis in Q, and that diameter of the circle which passes through Q meet the tangent in P; show that PR = B C. (See fig. Prop. XV.) 63. If P G the normal at P cut the major axis in G, and if DR, PN be the ordinates of D and P, prove that the triangles PGN, DRC are similar; and thence deduce that PG bears a constant ratio to CD. 64. The tangent at a point P of an ellipse meets the tangents at the vertices in V, V'. On VV' as diameter, a circle is described which intersects the ellipse in Q, Q'; show that the ordinate of Q is to the ordinate of P as B C to B C + CD; where CD is conjugate to OP. 65. PCP' is any diameter of an ellipse; the tangents at any two points E and E' intersect in F; PE', P'E intersect in G. Show that FG is parallel to the diameter conjugate to PCP'. K2 132 CONIC SECTIONS. 66. If P be any point on an ellipse, and with P as centre and the semi-axis minor as radius a circle be described; prove that if PG be the normal, a circle described on CG as diameter will cut the first circle at right angles. 67. ABC is an isosceles triangle having AB = AC. BD, BE drawn on opposite sides of B C, and equally inclined to it, meet A C in D and E. If an ellipse be described about BCD having its minor axis parallel to BC; then AB will be a tangent to the ellipse. 68. If A Q be drawn from one of the vertices of an ellipse perpendicular to the tangent at any point P; prove that the locus of the point of intersection of PS and QA produced will be a circle. 69. If Y, Y' be the feet of the perpendiculars from the foci of an ellipse on the tangent at P; prove that the circle circumscribed about the triangle YNY' will pass through C. 70. Prove that the angle between the tangents to the auxiliary circle at Y, Y', is the supplement of the angle SPS'. 71. P is any point on an ellipse; PMl, PN perpendicular to the axes meet respectively, when produced, the circles described on the axes as diameter in the points Q, Q'. Show that QQ' passes through the centre. 72. Assuming that the greatest triangle which can be inscribed in a circle is equilateral, prove by the method of projection, that the greatest triangle which can be inscribed in an ellipse has one of its sides bisected by a diameter of the ellipse, and the others cut in points of bisection by the conjugate diameter. CONIC SECTIONS. 133 73. PQ is a chord of an ellipse, normal at P, LCL' the diameter bisecting it. Show that PQ bisects the angle LPL', and that LP + L'P is constant. 74. A tangent to an ellipse at a point P intersects a fixed tangent in T; if through the focus a line be drawn perpendicular to ST meeting the tangent to P in Q; show that the locus of Q is a straight line touching the ellipse. 75. In an ellipse if a line be drawn through the focus making a constant angle with the tangent; prove that the locus of the point of intersection with the tangent is a circle. 76. Any chord PP' of an ellipse is produced to a point Q, such that P'Q is equal to half the diameter parallel to PP', and QB' R is drawn through the centre to meet the ellipse in BR, R'; show that the area PCR is three times the area PCR. 77. In an ellipse, L is the extremity of the latus rectum, and CD conjugate to CL. If a circle be described with centre C and passing through B, and a line be drawn through D parallel to the major axis, the portion of this line which lies within the circle will be equal to the latus rectum. 78. If P be any point in an ellipse, and K the point in which a normal at P intersects a line at right angles to it through S', E the point of intersection of SP, and the diameter conjugate to CP, and if EK and CK be joined, each of the figures S CKE, ' CEK will be a parallelogram. 79. If T be a point on the axis A A' produced, and PN the ordinate of the point where the tangent from T touches the ellipse; prove that AN. AN: AT. A'T:: CN: CT. 80. Given in an ellipse a focus and two tangents; prove that the locus of the other focus is a straight line. 134 CONIC SECTIONS. 81. A focus, a tangent, and the axis major being given, prove that the locus of the other focus is a circle. 82. A focus, a tangent, and the axis minor being given, prove that the locus of the other focus is a straight line. 83. An ellipse touches a fixed ellipse and has a common focus with it; if the major axis be fixed, the locus of the other focus is a circle; if the minor axis be fixed, the locus is an ellipse. 84. An ellipse and a parabola have a common focus. Prove that the ellipse either intersects the parabola in two points, and has two common tangents with it, or else does not cut it. 85. If in the ellipse a focus, a point, and the axis minor be given, the locus of the other focus is a parabola. 86. If at the extremities P, Q of any two diameters CP, CQ of an ellipse, two tangents Pp, Qp be drawn cutting each other in T, and the diameter produced in p and q, then the areas of the triangles TQp, TPq are equal. 87. If a straight line CN be drawn from the centre to bisect that chord of the circle of curvature at any point P of an ellipse, which is common to the ellipse and circle, and if it be produced to cut the ellipse in Q, and the tangent in T; prove that OP = CQ, and that each is a mean proportional between CN and CT. 88. An ellipse is described so as to touch the three sides of a triangle; prove that if one of its foci move along the circumference of a circle passing through two of the angular points of the triangle, the other will move along the circumference of another circle, passing through the same two angular points. Prove also that if one of these circles pass through the centre of the circle inscribed in the triangle, the two circles will coincide. 89. A triangle is described about an ellipse, so that the extremities of one of its sides lie in an ellipse, confocal with the given one; prove that the line bisecting the opposite angle passes through the pole of that side with respect to the outer ellipse. CONIC SECTIONS. 135 90. Prove the following construction for a pair of tangents from any external point T to an ellipse of which the centre is C. Join CT; let TPCP'T, a similar and similarly situated ellipse, be drawn, of which CT is a diameter, and P, P' its points of intersection with the given ellipse; TP, TP' will be tangents to the given ellipse. 91. The locus of the foci of all ellipses inscribed in the same parallelogram is a rectangular hyperbola. Prove this, and give a geometrical construction for finding the asymptotes. 92. AC is a fixed diameter of a circle, ABCD a quadrilateral figure inscribed in the circle; prove that if the angles BA C, DA be complementary, the locus of the intersection of BA, CD will be an hyperbola. 93. Prove that a circle can be described so as to touch the four straight lines drawn from the foci of an hyperbola to any two points on the same branch of the curve. 94. Any three diameters of an ellipse LL', MM', NN', being taken, a circumscribing parallelogram T 'UV touches the ellipse at L, L' M, M'. Show that a conic section can be described through the points R, T, U, V, N, N', which will be an hyperbola whose asymptotes are the lines forming in the ellipse the diameters conjugate to ATN' and to the other common chord of the ellipse and hyperbola. 95. On opposite angles of any chord of a rectangular hyperbola are described equal segments of circles; show that the four points in which the circles to which these segments belong again meet the hyperbola, are the angular points of a parallelogram. 96. A triangle is inscribed in a rectangular hyperbola: prove that the circle described through the middle points of the sides of the triangle passes through the centre of the hyperbola. 136 CONIC SECTIONS. 97. ACB is an isosceles triangle; AB the base, and D any point in CB or CB produced: if BZ be drawn parallel to AD, meeting CA or CA produced in Z, prove that the middle point of DZ will be in an hyperbola whose asymptotes are CA, CB. 98. An ellipse and hyperbola are described so that the foci of each are at the extremities of the transverse axis of the other; prove that the tangents at their points of intersection meet the conjugate axis in points equidistant from the centre. 99. In a rectangular hyperbola, PK, PL are drawn at right angles to A'P, AP respectively to meet the transverse axis in K and L; prove that PK is equal to AP and KL to AA', and the normal at P bisects KL. 100. In a rectangular hyperbola PC is a fixed diameter, Q any point on the curve; show that the angles QPO, QOP differ by a constant angle. 101. If the tangent at any point P of an hyperbola cut an asymptote in T, and if SP cut the same asymptote in Q, when SQ = QT. 102. If a given point be the focus of any hyperbola, passing through a given point and touching a given straight line, prove that the locus of the other focus is an arc of a fixed hyperbola. 103. At any P of an hyperbola a tangent is drawn, and PQ is taken on it in a constant ratio to CD; prove that the locus of Q is an hyperbola. 104. In an hyperbola, supposing the two asymptotes and one point of the curve be given in position, show how to construct the curve; and find the position of the foci. 105. If A, D be two fixed points, and the angle PAD always exceed PDA by a given angle; find the locus of P, and the position of the transverse axis and asymptote. CONIC SECTIONS. 137 106. From the middle point D of the base AB of the triangle ABC a straight line EDE' is drawn, making a given angle with AB, and the points E, E' are taken so that ED = E'D = 2 AB. If CA, CB take all possible positions consistent with the condition that the difference of the angles CAB, CBA is equal to EDA; prove that the point C will trace out a rectangular hyperbola of which AB, E'E are conjugate diameters. 107. In the rectangular hyperbola, prove that the triangle, formed by the tangent at any point and its intercepts on the axes, is similar to the triangle formed by the straight line joining that point with the centre, and the abscissa and semiordinate of the point. 108. Tangents are drawn to an hyperbola, and the portion of each tangent intercepted by the asymptotes is divided in a constant ratio; prove that the locus of the point of section is an hyperbola. 109. Show that the point of trisection of a series of conterminous circular arcs lie on branches of two hyperbolas, and determine the distance between their centres. 110. From a point R on one asymptote RE is drawn touching the hyperbola in E, and ET, EV are drawn through E, parallel to the asymptotes, cutting a diameter in T and V; R Vis joined, catting the hyperbola in P, p: show that TP, Tp touch the hyperbola. 111. Given in the ellipse a focus and two points, the locus of the other focus is an hyperbola. 112. If a rectangular hyperbola passes through three given points, the locus of its centre is a circle, which passes through the middle points of the lines joining the three given points. 113. If the tangent at P meet one asymptote in T, and a line TQ be drawn parallel to the other asymptote to meet the curve in Q; prove that if PQ be joined and produced both ways to meet the asymptotes in R and R', RR' will be trisected at the points P and Q. 138 CONIC SECTIONS. 114. If two concentric rectangular hyperbolas have a common tangent, the lines joining their points of intersection to their respective points of contact with the common tangent will subtend equal angles at their common centres. 115. If TP, TQ be two tangents drawn from any point T to touch a conic in P and Q, and if S and S' be the foci, then ST: S'Tl:: SP. SQ: S'P. S'Q. 116. The circle of curvature at the vertex of a conic meets the axis again in D, and a tangent is drawn to the circle at D: if two tangents be drawn to the circle from any point in the conic they will intercept between them a constant length of the former tangent. 117. If the lines which bisect the angles between pairs of tangents to an ellipse be parallel to a fixed straight line, prove that the locus of the points of intersection of the tangents will be a rectangular hyperbola. 118. An hyperbola, of given eccentricity, always passes through two given points; if one of its asymptotes always pass through a third given point in the same straight line with these, prove that the locus of the centre of the hyperbola will be a circle. 119. A, P and B, Q are points taken respectively in two parallel straight lines, A and B being fixed, and P, Q variable. Prove that if the rectangle APBQ be constant, the line PQ will always touch a fixed ellipse or a fixed hyperbola, according as P and Q are on the same or opposite sides of AB. 120. If two plane sections of a right cone be taken, having the same directrix, the foci corresponding to that directrix lie on a straight line which passes through the vertex. 121. Give a geometrical construction by which a cone may be cut so that the section may be an ellipse of given eccentricity. CONIC SECTIONS. 139 122. Given a right cone and a point within it, there are but two sections which have this point for focus; and the planes of these sections make equal angles with the straight line joining the given point and the vertex of the cone. 123. If the curve formed by the intersection of any plane with a cone be projected upon a plane perpendicular to the axis, prove that the curve of projection will be a conic section, having its focus at the point in which the axis meets the plane of projection. 124. If F be the point where the major axis of an elliptic section meets the axis of the cone, and C be the centre of the section; prove that CF: CS:: AA': AO + A'O, O being the vertex of the cone. THE END. R. CLAY SONS, AND TAYLOR, PRINTERS, BREAD STREET HILL.