278 CONIC Now, as long as PP remains parallel to itself, QQ must remain parallel to itself, and therefore its middle point W lies on a fixed straight line, the diameter at right angles to QQ . Therefore V lies on a fixed straight line through C, since WM : VM = BC : AC. The tangents at the points where CV cuts the ellipse will be parallel to the chords PP which CV bisects. Definition. If CD be drawn parallel to the tangent at P, then CD is said to be conjugate to CP. Now it is evident that CD and CP will correspond to two radii at right angles in the circle on AA as diameter, and therefore if CD is conjugate to CP, CP will also be conjugate to CD. PROP. XV. If CP, CD (fig, 24) be semi-conjugate diameters, then CP 2 + CD 2 = SJ ain xed lies be hen adii e if ACTIONS C Q2 /CW 2 --CV 2 V 1 A, r 2 CQ R 2 . ros CR 2 -C0 2 ^ CR2 Therefore PO OP -CO 2 RO.OR _pTj 2 LK^ or PO.OP :RO. OR = CD 2 :C Draw QPN, Q DN common ordinates of the circle and the ellipse. Fig. 24. Fig. 25. Then CQ, CQ will be at right angles, and therefore the two triangles QCN, CQ N will be equal in all respects. .-. NC = Q N and QN=CN . = BC 2 (Prop. xii.). =AC 2 + BC 2 . It follows that if the tangent at P meets the axes in T, T (fig. 25), then PT.PT = CD 2 . Draw ordinates PM, DM to the major axis, and PN to the minor axis. Then PT : PM-CD : DM and PT : PN = CD : CM .-. PT. PT : PM. PN = CD 2 : CM . DM . .-. PT . PT : CD 2 = PM . PN : CM . DM . _PM .DM CM CM = BC BO , . i AC AC .-. PT. PT V =CD 2 . PROP. XVI. Jf CP, CD be semi-conjugates, and QV be an ordinate parallel to CD, then QV 2 : PV . VP = CD 2 : CP 2 . Draw QR an ordinate parallel to CP, and. draw TJQW the tangent to the ellipse at Q, meeting CP, CD in U, W. Then CR.CW = CD 2 .-. CR 2 :CD 2 = CR:CW = UV:CU. Again CU.CV = CP 2 .-. CU:CV = CP 2 :CV 2 . . . CU - CV : CU = CP 2 - CV 2 : CP* or UV:CIT=PV.VP :CP 2 . Hence CR 2 :CD 2 = PV. VP rCP 2 or Q V 2 : P V . VP = CD 2 : CP 2 . PROP. XVIL If POP (fig. 26) be any chord, and ROCR the diameter through 0, then PO . OP : RO . OR - CD 2 : CR 2 , whero CD is the semi-diameter parallel to PP . Draw CVWQ conjugate to PP , meeting the curve in Q, and the ordinate through R in W. Fig. 26. Then TV 2 : CQ 2 - C V 2 = CD 2 : CQ 2 and RW 2 : CQ 2 - CW 2 = CD 2 : CQ 3 (Prop. Z vi. ) PV 2 - RW 2 . ~ : CQ 2 - C V 2 - (CQ 2 - CW 2 ) Now PV2 _ 2 . = PV 2 - OV PO . OP PROP. XVIII. If POP , pOp be any two chords, and CD, Cd the semi-diameters parallel to them, then PO.OP :pO.Op = CD 2 :Cd 2 . From the last proposition we have PO . OP : RO . OR = CD 2 : CR 2 and pO . Op = RO . OR = Cd 2 : CR 2 . Therefore PO . OP : CD 2 = RO . OR : CR 2 = pO.Op :Cd 2 or PO . OF : pO . Op - CD 2 : Cd 2 . PROP. XIX. If the two extremities of a rod slide along two fixed straight lines at right angles to one another, any fixed point in the rod will describe an ellipse. Let OM, ON (fig. 27) be the two fixed straight lines, and MPN any position of the rod, and P the tracing point. Complete the rectangle QMON and join OQ, and draw, parallel to ON, RPH to meet OQ in R and OM in H. Then it can easily be shown that OR = NP and that RH:PH = OR:PM = PN:PM. Tne locus of R is a circle whose centre is O and radius PN. H Fig. 27. And the locus therefore of P is an ellipse whose axes are in OM and ON and equal to PN, PM respectively. PROP. XX. If a circle roll on the inside of a fixed circle of double the radius, any fixed point in the circum ference of the moving circle will trace out a diameter of the fixed circle, and any other point in the plane of the moving circle will trace out an ellipse. If the point M (fig. 28) coincided with A at the beginning of the motion and the circle now touch at Q, the arcs MQ, QA must be equal. There fore if C and O be the centres, the angle QCM is double the angle QOM, Fig 28. and therefore OCQ is always a straight lire ; as also MCN. It is clear therefore that the motion of a point P in MN is exactly the same as in that of a point in the moving rod. (Prop, xix.) PART III. THE HYPERBOLA. DEFINITIONS. A straight line passing through the centre, and termin ated by the hyperbola, is called a diameter. The extremities of a diameter are called its vertices. The diameter which passes through the foci is called the transverse axis. A straight line BOB passing through the centre, per pendicular to the transverse axis, such that EC 2 = B C 2 = SC 2 - AC 2 is called the conjugate axis. Any straight line terminated both ways by the hyper bola, and bisected by a diameter produced, is called an ordinatt to that diameter. Each of the segments of a transverse diameter produced, intercepted by its vertices and an ordinate, is called an abscissa. A chord, tangent, and normal are defined exactly in the
sarre words as in the case of the parabola.