i)4 BOOK OF MATHEMATICAL PROBLEMS. 



460. If three tangents to a conic be such that their points of 

 intersection are at equal distances from one of the foci, each 

 distance will be equal to the major axis : and the second focus 

 will be the centre of perpendiculars of the triangle formed by the 

 tangents. 



461. A straight line is drawn touching the circle on the 

 minor axis of an ellipse, meeting the ellipse in P, and the director 

 circle of the ellipse in Q, Q'; prove that the focal distances of P 



are equal to QP, Q'P. 



462. A conic is inscribed in a triangle and is concentric with 

 the Nine Points' Circle ; prove that it will have double contact 

 with the Nine Points' Circle. 



463. EF is a chord *of a circle, S is its middle point ; con- 

 struct a conic of which E is one point, S one focus, and the given 

 circle the circle of curvature at E. 



464. If P be a point on an ellipse equidistant from the minor 

 axis and from one of the directrices, the circle of curvature at P 

 will pass through one of the foci. 



465. If S be the focus of a conic, K the foot of the directrix, 

 Q a point on the tangent at P, QR, QK perpendiculars on SP, 

 SK respectively; then will SR bear to KK a constant ratio. 



466. If an equilateral triangle PQR be inscribed in the 

 auxiliary circle of an ellipse, and P f 1 @, R' be the corresponding 

 points on the ellipse, the circles of curvature at P f t Q, Rf meet 

 in one point lying on the ellipse and on the circle circumscribing 

 PQ'K. 



467. From a point on an ellipse perpendiculars are drawn to 

 the axes and produced to meet the circles on these axes re- 

 spectively ; prove that the line joining the points of intersection 

 passes through the centre. 



468. TP, TQ are tangents to a conic, .Qrj, Pp chords parallel 

 to TP, TQ respectively; prove that pq is parallel to PQ. 



