(Art. 15.) to the intercepts on a line drawn from any point in the curve parallel to an asymptote made by parallel lines in any direction. 13. If the conjugate axis of an hyperbola is greater than the transverse, we may project it into a rectangular hyperbola having its axes equal to the transverse, and the same results will obtain as in the last Article, except that the angles between the asymptotes that contain the transverse axis will be obtuse instead of acute. 14. Supplemental Chords. P If any point Q in an hyperbola be joined with the extremities PP' of any diameter, PQ, P'Q are called supplemental chords, and are parallel to conjugate diameters. For if we take O the middle point of PQ and join CO, PQ will be parallel to the conjugate diameter to CO, but C is the middle point of PP' and therefore CO is parallel to PQ, hence PQ, P'Q are parallel to conjugate diameters. 15. The ratio of the rectangles of two intersecting chords. of an hyperbola is the same when one or each is moved into any position parallel to its former position. Let VP, Vp be the generating lines of the cone through the extremities of one of the chords POP, O being the point of intersection of the two chords. Through V the vertex draw VC parallel to POP of some fixed length, the same for all the chords. Let planes through O and Cat right angles to the axis of the cone intersect the plane PVCp (the plane of the paper) in the lines ROr, DdC; and the generating lines VP, Vp in R, r, D, d; then Rr, Dd are chords of the circles in which these planes cut the cone. Also Ror is parallel to DdC as well as POp to VC. Hence by similar triangles POR, VCD, PO RO VC: CD, and by similar triangles pOr, VCd, therefore PO. Op: RO. Or :: VC2: CD. Cd. Now let P'Op' be another chord of the hyperbola with its extremities on different generating lines of the cone, intersecting the former chord in O. Draw VC' parallel to P'Op' and of the same length as VC: also R' Or', D'd' C' the intersections of planes through O and C'at right angles to the axis with the plane PVC'p'. Then R'Or' is a chord of the same circle as ROr was, and RO.Or=R'O.Or', also VC VC: and as before P0. Op': R'O. Or' :: VC": C'D'. C'd'. Hence we shall have PO.Op: P'0. Op' :: C'D'. C'd' : CD. Cd. Now observe that if the chords Pop, P'op' are moved parallel to themselves into some new position, but so as to be still chords of the hyperbola, the lines VC, VC' will not be affected, but the generating lines on which the extremities of the chords rest will not be the same as before; yet C, C' remaining fixed, Dd, D'd' in their new position will be still chords of the same circles as before, and each of the rectangles CD. Cd, C'D'. C'd' will retain the same value as before. Hence the ratio PO.Op: P'O. Op' will be invariable, and have the same value wherever O may be situated, provided the chords are drawn always parallel to their original position. The same proof will hold when the extremities of the chords are on different branches of the curve, and when O is on the convex side of either branch. The same proof will also hold for the other sections of the cone equally with the hyperbola. 16. The ratio of the rectangles on the segments of the chords equals that of the squares on the parallel semidiameters. Let the chords move till they become tangents to the curve, viz. the tangents OP, OP' intersecting in O. The ratio of the rectangles is that of the squares on OP, OP'. Draw CQ parallel to OP to meet the conjugate in Q, and draw QQ parallel to PP' to meet CQ' parallel to OP in Q. Join CO, and let it meet QQ' in W, and when produced let it meet PP in V. Then V is the middle point of PP', and the triangles CWQ, CWQ' are similar to OVP, OVP' : and .. QW: WC :: PV : VO, .. QW: Q'W :: PV : P'V, but PV=PV; therefore also QW= Q'W. But CV bisecting PP', bisects all chords of the hyperbola and its conjugate parallel to PP', and therefore QQ', parallel to PP' and bisected by CO, is a chord of the conjugate hyperbola; C'Q' is the semi-diameter parallel to OP', and the ratio of the rectangles under the segments of the chords = OP OP' CQ: CQ, since the triangles OPP', OQQ' are similar. it = See also Besant's Elementary Conic Sections, p. 116. COR. If a circle intersect an hyperbola in four points, be easily shewn that the diameters parallel to the may pairs of opposite sides of the quadrilateral formed by drawing the common chords will be equal, and therefore equally inclined to the axes of the curve; hence the opposite sides of the quadrilateral will be equally inclined to the axes. 17. NG NC: BC: AC2. : Let PG be the normal at P; draw NR from the foot of the ordinate PN to touch the auxiliary circle at R; then if PT be the tangent at P, RT will be at right angles to the transverse axis, because CN. CT CA2. Also and = NR2NA. NA' and .. NP2: NR2 :: BC: AC2, Join CR: then in the right-angled triangles TPG, CRN, PN TN. NG, RN2 CN. TN, .. PN: RN2 :: NG; NC; = 18. = .. NG: NC:: NP2: NR2: BC2: AC2. PF.PG= BC2, PF. Pg=AC2. If the normal at Pintersects the axes in G and g, and PF be the perpendicular on CD the semi-diameter conjugate to CP, then first, PF. Pg = AC2. J. C. S. 5 |