. Plane and solid analytic geometry; an elementary textbook. b2. (See Chap. 14,Prob. 1.) 11. The line joining any point on the directrix of a pa- Ch. X, § 78] TANGENTS 139 rabola to the focus is perpendicular to the chord of contactof tangents from the point. Take the coordinates of the point L (Fig. 74) on the directrix as ( , yx), and show that the line LF which joins this point to the focus is perpendicular to the chord of contact yxy — mx ——- A 12. Prove the same theorem for the central conies. 13. The two tangents which may he drawn from anexterior point to any conic subtend equal angles
. Plane and solid analytic geometry; an elementary textbook. b2. (See Chap. 14,Prob. 1.) 11. The line joining any point on the directrix of a pa- Ch. X, § 78] TANGENTS 139 rabola to the focus is perpendicular to the chord of contactof tangents from the point. Take the coordinates of the point L (Fig. 74) on the directrix as ( , yx), and show that the line LF which joins this point to the focus is perpendicular to the chord of contact yxy — mx ——- A 12. Prove the same theorem for the central conies. 13. The two tangents which may he drawn from anexterior point to any conic subtend equal angles at the focus. 14. In the parabola the perpendicular from the focus onany tangent meets it on the tangent at the vertex; the per-pendicular meets the directrix on the line through the pointparallel to the axis of the parabola. The equation of thetangent at any point (xv yxy = mx + mxvThe equation of a per-pendicular to the tangentthrough the focus is my,yxx + my = -^l. The coordinates of thepoint of intersection ofthese two lines are . Fig. 75. They therefore meet on the l^-axis, which is the tan-gent at the vertex. Let the student prove the second part of the theorem. 140 ANALYTIC GEOMETRY [Ch. X, § 78 15. Show that theorem 14 does not hold for centralconies, but that the perpendiculars from the foci of a cen-tral conic on any tangent meet the tangent on the circlex2 + f = a2m (See chap> 14? Prob# 5^ 16. The perpendicular from a focus on any tangent toa central conic meets the corresponding directrix on the linejoining the centre to the point of contact of the tangent. 17. In any conic, tangents at the ends of the latus rectummeet the X-axis on the directrix. 18. The tangent at any point of the parabola meets thedirectrix and latus rectum produced at points equally dis-tant from the focus. 19. The product of the perpendiculars from the fociof a central conic on any tangent is constant and equalto b\ 20. The semi-minor axis b of a central conic is a meanproportional b
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