CONIC SECTIONS 277 PROP. V. The product of the perpendiculars from the foci on the tangent is constant. If SY, S Y (fig. 18) be the perpendicu- Jars from the foci on any tangent, then it is easily seen that, if YS be produced to meet the circle on AA as diameter again in Z, ZCY is a straight line, and S Y = SZ. Therefore S Y . S Y - SY . SZ - AS . SA (Eucl. iii. 35). PROP. VI. The tangent at any point of an ellipse makes equal angles with the focal distances of the point. Let the tangent ZPZ (fig. 19) at the point P meet the two direc- trices in Z,Z . Join ZS, SP, PS , z S Z , and draw MPM parallel to the axis, to meet the directrices in M, M . Then because SP : PZ = cPM : PZ = *PM : PZ = S P : PZ , and the angles PSZ, PS Z are right angles (Prop, vii.), there fore the triangles PSZ, PS Z are similar (Eucl. vi. 7). Therefore the angle SPZ = angle S PZ , Fig. 18. Fig. 19. PROP. VII. To draw a tangent to an ellipse at a point on the curve. First Metlwd. Join SP, S P, and draw a line bisecting the ex ternal angle of S PS . This line is the tangent at P. (Prop, vi.) Sc-cond Method. Draw SZ at right angles to SP, meeting the corresponding directrix in Z. ZP is the tangent at P. (Prop, iii.) Third Method. On SP as diameter describe a circle which will touch the circle on A A as diameter in a point Y. YP is the tangent at P. (Prop, iv.) PROP. VIII. To draw a pair of tangents to an ellipse from an external point. First Method. Let (fig. 20) be the point, and S, S the foci. Join OS. With centre and radius OS describe a circle ; and with centre S and radius equal to A A describe another circle. It can be shown that these two circles will always intersect in two points M, M . Join S M, S M , cutting the curve in P, P . Then OP, OP will be tangents to the curve. Join SP, SP . Now SP + PS ~AA = MS therefore SP-MP; and OS = OM ; therefore the two triangles OPS, 0PM are equal in all respects, and the angle PS = angle 0PM. Therefore OP is a tangent to the ellipse at P (Prop, vi.) Second Method. Let (fig. 21) be the o point, and S, S the foci. Join OS, and upon it as diameter de scribe a circle, cutting the circle described on AA as diameter (which it will always do) in Y, Y . Join OY, OY , and produce them if ne cessary to meet the curve in P, P . They will be tangents to the curve at P, P . Because OYS is a semicircle, the angle OYS is a right angle, and therefore OY is a tangent to the ellipse (Prop, iv.) Fig. 21, PROP. IX. If OP, OP be tangents to the ellipse at P, P , and S be a focus, the angles OSP, OSP are equal. In fig. 20 we have angle OSP = angle OMP = angle M MS - angle M MO -angle MM S -angle MM O = angle OM S = angle OSP . PROP. X. If OP, Or be two tangents to the ellipse, and S, S be the foci, the angles SOP, S OF are equal. Fig. 20. In fig. 21 suppose S Z, S Z be drawn perpendicular to OP, OP respectively, then SY. S Z -SY . S Z (Prop, v.) therefore SY : SY = S Z : S Z. Also the angle YSY = supplement of angle POP - Z SZ. Therefore the triangles YSY , Z SZ are similar, and the ancrle Y YS = S Z Z, and angle Y YS = angle Y OS, and angle S Z Z -angle S OZ. Therefore angle SOP = angle S OP . PROP. XL If C (fig. 22) be the middle point of A A , then CA* = CS, CX. SA :A X=c:l-SA : AX. . . SA + SA :SA-A X+ AX: AX, or AA :SA = XX :AX . . AA :XX = SA : . Again, SA -SA :SA = A X-AX :AX. or SS :SA = AA : AX .-. SS :AA = SA :AX=e:l. From (1) and (2), AA :XX = SS : AA , or CA:CX = CS:CA . . CA 2 = CS.CX. . Also CS :CX = CS 2 :CA 2 = CA 2 : CX 2 . (1) (3) (4) PROP. XII. I PN be an ordinate of the ellipse, then PN 2 always bears a con stant ratio to AN. NA . By similar triangles PNA , Z XA (fig. 22), PN : NA -Z X : XA , and by similar triangles PNA, ZXA, PN :NA=ZX :XA. Therefore PN 2 : AN . NA = ZX . XZ : XA . XA . Now it appears from Prop. iii. that the angle Z SZ is a right angle ; therefore ZX . XZ =SX 2 (Eucl. vi. 8.) ; and XA.XA = CX*-CA 2 . Therefore N 2 : AN . NA = SX 2 : CX 2 - CA 2 = CB 2 : CA 2 (Prop, ii.) PROP. XIII. The ordinates of the ellipse and of the circle described on AA as diameter are in a constant ratio. If, in fig. 22, NP be produced to meet the circle on AA as diameter in Q, then QN 2 =AN . NA and PN 2 ; AN . N A = CB : CA 2 (Prop, xii.) .-. PN:QN S = CB* : CA 2 andPN : QN = CB : CA. COROLLARY. The ordinates of two ellipses which have a common major axis are in a constant ratio. It can easily be shown from the last result, if QPN, Q P N be two common ordinates of the circle and ellipse, (1) that the chords PP , QQ will meet the axis in the same point ; (2) that the tangents at P, Q will meet the axis at the same Fi S 22. point ; (3) that the intersection of the tangents at P, P and the intersection of the tangents at Q, Q will lie on a straight line perpendicular to the axis ; (4) that the middle points of PP and 3Q lie on a straight line perpendicular to the axis. It can also be shown by means of this proposition that the area of the circle is to the area of the ellipse as AC to BC, and that the area of the parallelogram formed by the four tangents at the ex tremities of two conjugate diameters (see definition below) is con stant, and is equal to AC . BC. PROP. XIV. The middle points of all parallel chords in an ellipse lie on a straight line through the centre. Let QPN, Q P N (fig. 23) be two common ordi nates of the ci i cle on A A as liameter and the ellipse. Let W, V be the middle points of QQ , PP - W, V lie on a straight line which
bisects NX at right angles. Fi.sr. 23.