CONIC SECTIONS 281 and from prop. xii. . PN 2 :AN.NA -BC a : AC 2 . . . CXN 2 -PN 2 : CN 2 -AN.NA =BC 2 : AC 2 but QN 2 -PN 2 = QP .QP, and AN.N"A = CN 2 -CA 2 ; .-. QP.QP : CA 2 = BC 2 : CA 2 , QP.QP =BC 2 . It is easily seen tbat QP-Q F, therefore PQ.PQ = BC 2 . Similarly it can be shewn that if RQR be drawn parallel to AA , to meet the hyperbola in R, R , then QR.QR = AC 2 . It is clear that the further the point Q moves away the greater the line PQ becomes, and it can be made greater than any assign able quantity, however large; and since PQ.PQ = BC 2 , therefore the line PQ becomes smaller and smaller, and can be made less than any assignable quantity, however small. Hence the asymptote never actually reaches the curve, though the distance between them con stantly decreases, and can be made smaller than any assignable quantity. It can easily be shewn that if the asymptote cuts a directrix in the point F, then CF=CA . As the asymptote may be considered as the tangent to the hyper bola at a point at an infinite distance, the foot of the perpendicular from the focus on the asymptote must lie on the circle whose diameter is A A (Prop, iv.) SF therefore must be perpendicular to the asymptote, as appears from other reasons (from Prop, iii., for example. ) Prop. XV. If QPP Q (fig. 37) be any chord cutting the asymptotes in Q, Q and the curve in P, P , then QP = P Q . and QP. PQ = CD S , where CD is the semi-diameter in the conjugate hvperbola parallel to PP . Draw RPR , DWW perpendicular to the transverse axis, meet ing the asymptotes in R, R , and W. W. Fig. 37. Then from similar triangles PRQ, DWG PQ : PR = CD :DW and from similar triangles PR Q , DWC PQ : PR = DC : DW. Therefore PQ . PQ : PR, PR = CD 2 : DW . DW ; but D W . D W = BC? - PR . PR (Prop. xiv. ) therefore PQ.PQ = CD 2 = P Q. P Q . Now, if V be the middle point of QQ , then PQ.PQ = QV -PV 2 and P Q.P Q = QV 2 -P V 2 . Therefore PV = P V. Thus V is the middle point of PP , as well as of QQ , or in other words PQ = Q P . It is clear that when the points PP coincide, or we have the tangent parallel to PP , say q p q, then q 2 )=pq = CD, and also that the line Cp will bisect all chords parallel to the tangent at p. PV is called an ordinate to the diameter Cp. DEFINITION. A chord which is parallel to the tangent at P is said to be con jugate to CP. If a diameter CD be drawn parallel to the tangent at P to meet the conjugate hyperbola in D, CP, CD are said to be conjugate semi-diameters. It is clear from Prop. xv. that a diameter is conjugate to the chords which it bisects. PROP. XVL If CD be conjugate to CP, then CP is conjugate to CD. Let the tangent at P meet the asymptote in L, then PL is parallel to CD ; it is also equal to CD (Prop, xv.) ; therefore DL is equal and parallel to CP (Euclid i. 33). Therefore LD is the tangent to the conjugate hyperbola at D, and therefore CP is conjugate to CU. It is easily seen that PD is bisected by one asymptete, and parallel to the other asymptote. PROP. XVII. If P CPV (fig. 38) be a diameter, and QV be an ordinate to CP, then QV- : PV.P V=CD 2 : CP 2 . Draw PL the tangent at P to meet an asymptote in L, and let Q V produced meet the asymptotes in R, R . Then Therefore Fig. 38. 2 =RQ.QR =PL 2 . V 2 = RV 2 -PL" 2 CV 2 Therefore = CV 2 -CP_ 2 ~CP* PV P V 1 * x Y CP 2 QV 2 : PV.P V = CD 2 : PROP. XVIII. If POP (fig. 39) be any chord, and ORCR the diameter through 0, then PC . OP : RO . OR =CD ! : CR*, where CD is the semi-diameter parallel to FP . Draw CQWV conjugate to PP meeting ths curve in Q, and the ordiuate through R in W. Then Now and = PV 2 -RW 2 . PO.OP =PV 2 -OV 3 CV 2 CW 2 PV 2 ; CV 2 -CQ 2 = CD 2 : CQ 2 RW 2 :CW 2 -CQ 2 = CD 2 : CQ 2 . -. PV - RW 2 , 3 : CV 2 - CQ 2 - (C W 2 - CQ 5 ) ^ PO . OP : CQ 8 - l = CD 2 : CQ 2 . Therefore
- CQ 2
. . PO . OP : CD 2 = CV 2 - C W 2 : CW 2 PO^ R,^ * CR,^ = RO . OR : CR 2 .
PO . OP : RO . OR - CD 8 : CR 3 . VI. 36
