BOOK OF MATHEMATICAL PROBLEMS. 



If T lie on the straight line joining A to the point of contact 

 with BC, will coincide with A. 



869. A conic is described touching the sides of a triangle 

 ABC, one of them BC in a fixed point A' ; B f , C f are two other 

 fixed points on BC ; prove that the point of intersection of 

 tangents drawn from tf, ' to the conic lies on a fixed straight 

 line. 



870. The sides of a triangle which is self- conjugate to a 

 given rectangular hyperbola touch a parabola, and a diameter 

 of the hyperbola is drawn through the focus of the parabola; 

 prove that the conjugate diameter is parallel to the axis of the 

 parabola. 



871. TP, TQ are two tangents to a parabola; a hyperbola 

 described through T, P, Q, and having an asymptote parallel to 

 the axis of the parabola meets the parabola again in R; prove that 

 its other asymptote is parallel to the tangent to the parabola 

 at E. 



872. TP, TQ are two tangents to a hyperbola; another 

 hyperbola is described through T, P, Q with asymptotes parallel 

 to those of the former ; prove that it will pass through the centre 

 C of the former, and that GT will be a diameter. 



873. A triangle is self-conjugate to a conic, and from any 

 other two points conjugate to the conic tangents are drawn to a 

 conic inscribed in the triangle ; prove that the other four points 

 of intersection of these tangents will be two pairs of conjugate 

 points to the first conic. 



874. If a conic pass through four points, its asymptotes meet 

 the conic which is the locus of centres in two points at the 

 extremities of a diameter. 



875. Four points and a straight line being given, four conies 

 are described, such that with respect to any one of them three 

 of the points are the angular points of a self-conjugate triangle,. 



