Page:Encyclopædia Britannica, Ninth Edition, v. 6.djvu/303

Rh CONIC SECTIONS 275 centre and radius OS describe a circle cutting the directrix (wliicl it will always do) in M and M. Draw MP, M P, parallel to the axis, cutting the parabola in P am P. Join OP, OP. They are tangents to the parabola at P and P. Join SP, SP. Ill the triangles OPS, 0PM, OP, PM = OP, PS each to each, and Oil = OS by the construction ; therefore the angles OPS, 0PM are equal, and therefore OP is a tangent to the curve at P (Prop, iv.) Second Method. Let (fig. 9) be the point. Upon OS as diameter describe a circle cutting the tangent at the vertex (which it will always do) in Y and Y. Join YO, Y O, and, if necessarj produce them to meet the curve in. P, P. They will be tangents to the curve at P, P. Because OYS is a semicircle, therefore the angle OYS is a right angle, and therefore YO is a tan gent to the parabola (Prop, iv.) PROP. VII. If OP, OP (fig. 8) be tangents to the parabola at P, P, then the triangles OSP, P SO are similar, andS0 2 = SP. SP. Because the angle OSP=anglo OMP = angle OM P = angle OSP , and the angle PS = angle OPM= angle SMA1 = J angle SOM (Euel. lii. 20) = anglo SOP (Prop, vi.); therefore the remaining angles **S- - OPS, P OS are equal, and the triangles OPS, P OS similar; and therefore (Encl. vi. 4) SP : SO - SO : SP, or SO 3 = SP. SP. PROP. VIII. If be the intersection of OP, OP, the tangents to the parabola at P and P, then 0V drawn parallel to the axis will bisect PP. From fig. 8 we see that, if a line through meet MM in Z, MM is bisected in Z ; and, because MP. ZOV, and M P are parallel, therefore PP is bisected in. V. PROP. IX. The angle between two tangents is equal to half the angle sub tended at the focus by the chord of contact. From fig. 9 we see that anglo YOY = angle YSY = angla YSA - angle Y SA = anglo PSA -| angle P SA = auglo PSP. It may be shown, by means of this proposition, that the circle which is described about the triangle formed by any threo tangents to a parabola passes through the focus. PROP. X. If OV (fig. 10) meet the parabola in Q, the tangent at Q is paral lel to PP, and OV will be bisected in Q. Draw the tangent RQR at Q, meeting OP, OP in R, R. J i n PQ, and let RW be drawn paral lel to OQV meeting PQ in W Then PW=WQ (Prop, viii.) Therefore OR = RP (Eucl. vi. 2). Similarly we can shew that OR -R P. Therefore OR:RP=.OR : R P , and therefore RR is parallel to PP (Eucl. vi. 2) ; and also Fig. 9. Fig. 10. . . OQ = QV. PROP. XL If V be the middle point of a chord PP, and Q be the point a* which the tangent is parallel to PP , then PV 2 -=4SO. QV. Suppose in fig. 10 SQ, SR joined, and PO produced to meet the axis in T. Then angle OKQ-J angle QSP (Prop, ix.) -angle QSR (Prop, yii ), and angle QKS = angle RPS (Prop, vii.) Bangle STP (Prop, iv )= angle QOR (Eucl. i. 29) ; therefore the two triangles OQB. SQR are similar, and OQ:QR=QR:SQ therefore QR 2 = SQ. OQ. But P V = 2Q R, and OQ - QV. therefore P V 2 = 4SQ. Q V. PROP XII. The parameter of the diameter QV is 4SQ. If the tangent at Q (fig. 11) meets the axis in T, SQ = ST -QV. Therefore the equality PV 2 = 4SQ. QV T. becomes PV 2 = 4SQ 2 therefore PV = 2SQ or PP =4SQ. PROP. XIII. If POP (fig. 12) be any chord, and OR be drawn parallel to the axis through any point O to meet the curve in R, then PO. OP 4SQ. RO, where 4SQ is the parameter of diameter PP. Draw RW parallel to PP to meet QV in W. Then PV 2 = 4SQ.QV and RW 2 = 4SQ.QW, and RO = W V and RW = OV. Therefore PO. OP = PV 2 - OV 2 (Eucl. ii. 5). = PV 2 -RW 2 = 4SQ.QV-4SQ.QW = 4SQ. WV-4SQ.RO PROP. XIV. If POP, pOp beany two chords intersecting in 0, and Q, q are the points of contact of the tangents parallel to them, then PO.OF:pO.Op -SQ:fiq. By Prop, xiii., PO. OP = 4SQ. RO and similarly pO. Op = 4Sq. RO. Therefore PO. OF :pO. Op =4SQ. RO : 4Sq.RO = SQ : Sq PROP. XV. The area included between any chord of a parabola and the curve is two- thirds of the area of the triangle formed by the chord and the tangents to the curve at its extremities. It is easily seen in fig. 10 that the area of the triangle ORR is one quarter the area of the triangle OPP, and therefore one half the area of the triangle QPP. N&quot;ow if we draw tangents where RW, R W meet the curve, we shall have two pairs of triangles whose areas are in the ratio 1 : 2, and so we may go on indefinitely. The sum of all the external triangles will be half the sum of all the internal triangles. The sum of the external triangles is the curvilinear area OPQP^ and the sum of the internal triangles is the curvilinear area PQF. Therefore 2 x area OPQP =area PQP . . 2 x triangle OPP = 3 area PQF or area PQ P = $ triangle OPP. PART II. -THE ELLIPSE. DEFINITIONS. A straight line passing through the centre, and termi nated both ways by the ellipse, is called a diameter. The extremities of a diameter are called its vertices. The diameter which passes through the foci is called the ransverse axis, also the major axis. The diameter which is perpendicular to the transverse axis is called the conjugate axis, also the minor axis. Any straight line not passing through the centre, but erminated both ways by the ellipse, and bisected by a diameter, is called an ordinate to that diameter. Each of the segments of a diameter intercepted between Is vertices and an ordinate, is called r.n abscissa. Fig. 12.