132 BOOK OF MATHEMATICAL PROBLEMS. 



one of the rectangular hyperbolas xy = c 2 is also the locus of the 

 feet of the perpendiculars let fall from the origin on tangents 

 to the hyperbolas jc 2 -y a = 4c 2 . 



653. The axes of an ellipse are the asymptotes of a rectan- 

 gular hyperbola which does not meet it in real points ; prove that 

 if two tangents be drawn to the ellipse from a point on the hyper- 

 bola, the difference of the eccentric angles of the points of contact 

 will be least when the point lies on an equi- conjugate of the 

 ellipse. 



654. The locus of the equation 



=*+- - 



X + X + . . . tO 00 , 



is that part of the hyperbola y 2 x s = c 2 , which, starting from a 

 vertex, goes to oo on the line y = x. 



655. Normals are drawn to a rectangular hyperbola at the 

 extremities of a chord parallel to a given line ; the locus of their 

 intersection is another rectangular hyperbola whose asymptotes 

 make with the asymptotes of the given hyperbola angles equal 

 and opposite to those made by the given line. 



656. An ellipse is described confocal with a given hyperbola, 

 and the equi-conjugates of the ellipse lie on the asymptotes of the 

 hyperbola ; prove that if from any point of the ellipse two tan- 

 gents be drawn to the hyperbola, the centres of two of the circles 

 which touch these tangents and their chord of contact will lie on 

 the hyperbola. 



657. A triangle circumscribes a given circle, and its centre 

 of perpendiculars is a given point \ prove that its angular points 

 lie on a fixed conic which is an ellipse, parabola, or hyperbola, as 

 the given point lies within, upon, or without the given circle. 



