102 BOOK OF MATHEM ATICAL PROBLEMS. 



526. PP f is a diameter of a rectangular hyperbola, QQ ', RR 

 two ordinates to it on opposite branches ; prove that a common 

 tangent to the circles whose diameters are Qty, RR will subtend 

 a right angle at P and at P'. 



527. Circles are drawn through two given points, and diame- 

 ters drawn parallel to a given straight line ; prove that the locus 

 of the extremities of these diameters is a rectangular hyperbola 

 which asymptotes make equal angles with the line of centres of 

 the circles and with the given straight line. 



528. The tangent at a point P of a rectangular hyperbola, 

 and the diameter perpendicular to CP are drawn ; prove that the 

 segments of any other tangent, from the point of contact to the 

 points where it meets these two lines, will subtend supplementary 

 angles at P. 



529. The normal at a point P of a rectangular hyperbola 

 meets the curve again in Q : two chords PA', PR', are drawn 

 through P at right angles to each other ; prove that the points of 

 intersection of QR, PR, and of QR, PR lie on the diameter at 

 right angles to CP. 



