92 BOOK OP MATHEMATICAL PROBLEMS. 



444. A circle and parabola meet in four points and tangents 

 are drawn to the parabola at these points ; prove that the axis of 

 the parabola will bisect the three diagonals of the quadrilateral 

 formed by these tangents. 



II. Central Conies. 



445. If SY be perpendicular on the tangent, SZ on the 

 normal, S being the focus, YZ will pass through the centre. 



446. A common tangent is drawn to a conic and to the 

 circle whose diameter is a latus rectum; prove that the latus 

 rectum bisects the angle between the focal distances of the points 

 of contact. 



447. A perpendicular from the centre on the tangent meets 

 the focal distances of the point of contact in two points ; prove 

 that either of these points is at a constant distance from the feet 

 of the perpendiculars from the foci on the tangent. 



448. The tangent at a point P meets the major axis in T \ 

 prove that SP : ST :: AN -. AT, N being the foot of the ordinate 

 and A the nearer vertex. 



449. The circle passing through the feet of the perpendiculars 

 from the foci on the tangent and through the foot of the ordinate 

 will pass through the centre; and the aDgle subtended at either 

 extremity of the major axis by the distance between the feet of the 

 perpendiculars is equal or supplementary to the angle which either 

 focal distance makes with the corresponding perpendicular. 



450. A series of conies having a common focus S and major 

 axes equal and in the same straight line will all touch the two 

 parabolas having the same focus S, and latus rectum a line coinci- 

 dent with the major axes in direction and of double the length. 



451. A conic is described having the same focus as a parabola 

 and major axis coincident in direction with the latus rectum of 



