A first course in projective geometry . LS is the polar of L. .. SLL is a self-polar triangle, and SL, SL are a pair ofconjugate lines at S. L 162 PROJECTIVE GEOMETRY But these are orthogonal by the definition of a focus, andif ST cuts QQ at V, LQVQ is a harmonic range (Ch. VIII.§7, VI.). .*. SL, SL must be the internal and external bisectors ofthe angle QSQ. When the conic is a hyperbola and TQ and TQ are drawnto touch different branches, SL will be the external bisector,and accordingly the angles TSQ, TSQ will be supplementary. Cor. ST cuts the conic at the point of contact of thetangen
A first course in projective geometry . LS is the polar of L. .. SLL is a self-polar triangle, and SL, SL are a pair ofconjugate lines at S. L 162 PROJECTIVE GEOMETRY But these are orthogonal by the definition of a focus, andif ST cuts QQ at V, LQVQ is a harmonic range (Ch. VIII.§7, VI.). .*. SL, SL must be the internal and external bisectors ofthe angle QSQ. When the conic is a hyperbola and TQ and TQ are drawnto touch different branches, SL will be the external bisector,and accordingly the angles TSQ, TSQ will be supplementary. Cor. ST cuts the conic at the point of contact of thetangent from L. We have therefore The portion of a tangent intercepted between the curveand the directrix subtends a right angle at the focus. This also follows from the fact that the polar of a point onthe directrix passes through the focus. The equal angles subtended at the focus by tangents fromthis point are, accordingly, each right angles. § 2. The Tangent at any Point is equally inclined to theFocal Distances of its Point of Fig. 83. Taking the figure (83) for the ellipse, we have, by the lastcorollary, PSF is a right angle. TANGENT AND NORMAL PROPERTIES 163 Let PG be the normal at P, and KF the directrix correspond-ing to S. Then angle SPG = angle SFP, each being complementary to SPF, = angle SKP, since SFKP is cyclic. Also angle PSG = angle SPK. ,. the triangles SPG, SPK are similar, and SG_SP_SP~PK~^* Similarly dealing with S and its corresponding directrix, we shall find that -— = . SG S^GSP~SP .. PG bisects the angle SPS, and therefore the tangentalso is equally inclined to the focal distances. The figure for the hyperbola is easily drawn. Cor. L In the case of the parabola SG = SP, S is at infinity, and the corresponding focal distance is parallel tothe axis. We have then in the parabola: TJie tangent is equally inclined to the axis and to the focaldistance of its point of contact. Cor. 2. If an ellipse and hyperbola are confocal, havethe same foci,
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