6

CONIC SECTIONS.

t'.ianSIIR, a right angle; therefore, if GSP he made equal toSGP, SP, GI\ will meet somewhere in P, which is a point in thecurve. For draw PE perpendicular to the directrix, and theangle PGE being equal to CIID, PG is to PF. in the constantratio mentioned Prop. 11 ; therefore SP is to FE in the same ratio,and P is a point in the hyperbola.

Secondly, let gp be drawn parallel to the asymptote, at a greaterdistance from the focus. Join Sg, meeting 11C in i; the angleSgp = SiTl which is less than a right angle; if therefore gSp hemade equal to S gp, the lines Sp, gp, will meet when produced insome point p, which is in the opposite hyperbola; for the anglepge being equal to CHD, pg is to p e, or'SP is t ope in the sameconstant ratio; thereforep is a point in the curve.

Cor. Hence, if any line be drawn through the centre of an hy-perbola within the angle contained by the asymptotes, it will meetboth the curves.

Prop. XIX. Fig. 27.

The asymptotes are diagonals, of the rectangle, which is madeby drawing tangents through the vertices of the four hyperbolas.

’ Let the tangents GA a, 1M i, which are drawn through the ver-tices of the transverse axis, meet the asymptotes in G, a, and I, i.Join IB, GB, as also ah, i't. The triangles MCI, ACrt, are equal,for AC = C M, the angle MCI = AC a, and CMI = C An, there-fore MI = A a, which is equal to GP>, Prop. 17. In like mannerit may he proved that Mi = C/»=CB; therefore, Ill, BG, areequal and parallel to MC, CA, the angles IBC, GBC, are each aright angle, and IPG is one straight line, which is equal and paral-lel to MA. For the same reason il/a is one straight line, which isequtl and parallel to MA; and because the lines IBG, iba, areerpendicular to the axis BC h, they arctangents to the conjugateyper'oolas, and Kiwi iv a rectangle of which the asymptotes In,G i, are the diagonals.

Cor. 1 . The asymptotes GCt, ICn, are also asymptotes to theconjugate hyperbolas. For HI = BG = CA, which is the semi-coniugite axis to the hyperbolas I.BR, Ibr.

Cor. 2 . If the hyperbolas be equilateral, the asymptotes willbe perpendicular to each other. See Def. 25, 2d.

Prop. XX. Fig. 29, 30, 31.

If a straight line Vo, which cuts a conic section or opposite sec-tions in two points P, p, meets the directrix in II, and a rightline HST be drawn through the focus, and SP, S p, be joined; theangle PSII will be equal topST.

Draw p'T parallel to PS, and let it meet IIS in T; and drawPE, pe, perpendicular to the directrix. The triangles 11PE, 11 pr,are similar, as also USD, 1 l ip, and SP : PE : : Sp : pc; and al-ternately SP : Sp : : PE : pc : : HP : Up :: SP : pT; thereforeSp = Tp, and the angle pS P =pTS = PSII.

Cor. 1 . When P and p coincide, fig. 29, 31, or when IIP be-comes a tangent to the conic section, SP will coincide with Sp,and the angles PSII, pST, will be right angles.

Cor. 2. Hence if a line SP he drawn from the focus to anvpoint P in a conic sec ti, n, fig. 32, and SII he drawn perpendicu-lar to SP, meeting the directrix in II, and IIP be joined, it willtouch the conic section in P. '*

Cor. 3. It is evident from this proposit : on, that a straight linecannot cut a conic section in more points than two.

Trot. XXI. Fig. 32.

If a tangent be d''awn to any point in the parabola, it will bisectthe angle contained by the two straight lines drawn from the pointof contact, one to the'focus, and the other to the directrix.

Let PH, which touches the parabola at P, meet the directrix inII. Join SP, SII, and draw PE perpendicular to the directrix.The angle SPE is bisected by PH. For SP = PE, and PII iscommon to the triangles SPIl, EPII, and PSH, PEI1, are rightangles, therefore the'triangles SPII, EPH, are similar; and theangle SPII = EPII.

Cor. 1 . Hence if a straight line PH bisects the angle SPE, itwill be a tangent to the parabola at the point P.

Cor. 2 . Let PS meet the curve again in p, and let III bedrawn parallel to the axis, it will bisect Pp in I, and HI will he bi-sected by tlie curve in A. For the angle 111 P IIPE = HPI,therefore’IP = III, and if p II be joined, the line pll yvill be a

tangent, and therefore for the same reason Ip = 111, thereforeIl’=Ip. Secondly, because SA=AII, the angles ASH =AHS, and the complements of these angles are equal, that is, AS!= AIS, therefore AI = AS = AIL

PRor. XXII. Fig. 33, 34.

If a tangent he drawn to any point in an ellipse or an hyperbola,and two lines be drawn from the point of contact to the foci, (heangles contained by each of these lines and the tangent are equal.Let PT touch the ellipse or hyperbola at any point P, let it meetthe directrices in T and /. Through V draw a line parallel to theaxis AM meeting the directrices in E and r, draw PS, PH, to theloci, and join ST, lit. Because the triangles 1 PE, tVe, are simi-lar, PE : PT : : Pc : 17, and SP : PE :: HP : Pc, therefore SP:PT : : IIP : IV, and the angles PST, PIlZ, are right angles, Cor.

I, Prop. 20, therefore the triangles SPT, 1117 are similar, and theangle SPT = IIP/.

Prop. XXIII. Fig. 35, 36.

The tangents at the vertices of any diameter of an ellipse or anhyperbola are parallel.

Let PCG beany diameter of an ellipse or hyperbola; draw thetangents PQ, GK, and join SP, PII, SG, Gil. Because SC =CII, and CP = CG, and the angle SCP = GCII, also SP isequal and parallel to Gil, therefore PII is equal and parallel toSG, and SPI1G is a parallelogram; therefore the angle SPH -VSGH, and the halves of these angles, fig. 36, or the halves of theirsupplements, fig. 35, will he equal, that is, the angle SPQ =IiGR, and if these he added to the equal angles SPC, C'GII, inthe ellipse, and subtracted from them in the hyperbola, Cl’Q =CGR, therefore PQ is parallel to GR.

Prop. XXIV. Fig. 37, 33, 39.

If two straight lines Vp, Qq, which meet each other in anv pointL, and are inclined to the directrix at any given angles LUX,L/iX, cut a conic section, or opposite sections, in the points P, p,and Q, < 7 ; the rectangles Ll’x Lp, and LQxLiy, will he in aconstant ratio to each other.

Let S be the nearest focus. Join IIS, and produce it if neces-sary; also join SP, Sp, Draw LX, PE, perpendicular to the di-rectrix; and draw I.T, Lf, parallel to SP, S/j, meeting IIS in Tand/. Because (Prop. 20 ,) the angle PSH = pST, fig. 37, and39, and=pSW, fig. 38, the angle LTf = 1./T, and LT = L/.On L as a centre, at the distance I.T or I./, describe a circle cut-ting IIl’p in M and m. Join SL, and produce it to meet the cir-cle in D and d ; and because the triangles IIPE, F1LX, arc simi-lar, as also Ill’S, 11LT, LT : SP :: LI1 : PH : : LX : PE; andalternately LT : LX :: SP : PE, that is, in the constant ratiomentioned Prop. 11 ; therefore the radius of the circle is givenwhen the distance of L from the directrix is given, whatever bothe position of the line Vp. And because I.T is parallel to PS,and LZ to pS,

LP : TS : : I.II : I1T,and pL : S/ :: I.II : Zli, thereforePI.xI.p : TSxS/ : : LID : IITx/ILBut TSx S/ = DSx Sd, and TIIx 11/ = MlJx lint — I.II*—LM 3 , fig. 37, 38, or = LM*—I.II 3 , fig. 39. Therefore PLxLp : DSx Sd :: LID: LID—LM 3 , or LM 3 —LID. ButLID : LT 3 , or LM 3 :: PII 3 : l’S 3 , and by division LID: : LID— LM'*, or LM 3 —LH 3 :: I’ll 3 : I’ll 3 —PS 3 , or l’S 3 —PID; which ratio depends only upon the constant ratio, Prop.

II , and the angle LIIX, SP being to PH in a ratio which is com-pounded of the ratios of SP to PE, and PE to PII, or of the con-stant ratio, and of the sine of the angle LI1X to radius. In tin*same manner it may be proved that the rectangle QLx Lq is toDSx Sd in a ratio depending only on the same constant ratio, andthe angle L/iX, therefore PLx Lp is to QLxLr/ in a constantratio, whatever be the distance of L from the directrix.

Cor. 1. If either of the lines Vp, Q q, or both of them becometangents to the conic section, or opposite sections, the squares ofthe tangents must be substituted for the rectangles l’Lx l.p, QLxl.q. For let LP touch the section in 1’, fig. 40, 41, thenQL being parallel to SP, by the preceding Prop. LP : QS: :I.H : HQ ; and LP* : QS* :: LII 3 : HQ 3 ; hut QS 3 = DSxSd, and QID = Mil x flm = LID — I..M 3 , therefore LP* : DSxSd; ; MID : LID — LM 3 , which was proved to be a con-stant