CONIC SECTIONS, GEOMETRICAL. <W 



series of parallel straight lines the ratio QP. PQ' : RP* is con- 

 stant. 



502. S, S' are foci of an ellipse whose minor axis is equal to 

 SS', P any point on the ellipse, the centre of the circle circum- 



<1 to SPS' ; prove that the circle on OP as diameter will 

 touch the major axis at the foot of the normal at P. 



503. Through different points of a given straight line are 

 drawn chords of a given conic, bisected respectively at the points; 

 prove that they will touch a fixed parabola. 



.504. With a fixed point on a conic as focus is described a 

 parabola touching any pair of conjugate diameters of the conic ; 

 prove that this parabola will have a fixed tangent parallel to the 

 tangent at 0, and that this tangent divides CO in the ratio 

 C0* : C&*, CO, CO' being conjugate radii. 



505. Through a point are drawn two straight lines, each 

 passing through the pole of the other with respect to a _ 

 conic ; any tangent to the conic meets them in P, Q ; prove that 

 the other tangents drawn from P, Q to the conic intersect on the 

 polar of 0. 



506. A parabola is described having tli. focus S of a given 

 conic for its focus and touching the minor axis; prove that a 

 common tangent to the two curves will subtend a right angle at S, 

 and that its point of contact with either conir lies on the directrix 

 of the oti 



III. Rectangular Hyperbola. 



A, B, C, D are four points on a rectangular hyperbola 

 and nC is perpendicular to AD; prove that CA is perpendicular 

 to BD and AB to CD. 



508. The angle between two diameters of a rectangular 

 'la is equal to the angle between th> .jugate diameters. 



