CONIC SECTIONS, GEOMETRICAL, 101 



517. A parallelogram has its angular points on the arc of a 

 rectangular hyperbola and from any point on the hyperbola are 

 drawn two straight lines parallel to the sides ; prove that the four 

 points in which these straight lines meet the sides of the paralle- 

 logram lie on a circle. 



518. The tangent at a point P of a rectangular hyperbola 

 meets a diameter QCQ* in T ; prove that CQ, TQ subtend equal 

 angles at P. 



519. Through any point on a rectangular hyperbola are 

 drawn two chords at right angles to each other; prove that 

 the circle passing through the point and bisecting the chorda will 

 pass through the centre. 



520. If PG be a fixed diameter and Q any point on the 

 curve, the angles QPG, QGP will differ by a constant angle. 



521. A, B are two fixed points, P a point such that AP, ///' 

 make equal angles with a given straight line; prove that the 

 locus of P is a rectangular hyperbola, 



V# is any chord of a rectangular hyperbola, CP a radius 

 perpendicular to it ; prove that the distance of P from either 

 asymptote is a mean proportional between the distances of 

 from the other. 



523. Two circles touch the same branch of a rectangular 

 hyperbola in the points P, Q, and touch each other at the centre 

 ' ' : prove that the angle PCQ = 60*. 



524. On opposite sides of any chord of a rectangular hyper- 

 bola are described equal segments of circles ; prove that tli- 

 point* in which the completed circlet again meet the hyperbola 

 are the angular points of a parallelogram. 



525. A circle and rectangular hyperbola intersect in four 

 points ; prove that the diameter of the hyperbola which U per- 



ular to the chord joining any two of the pointa will bisect 

 the chord joining the other two. 



