100 BOOK OF MATHEMATICAL PROBLEMS. 



509. AA' is the transverse axis, P any point on the curve, 

 PK y PL are drawn at right angles to AP, A'P to meet the axis ; 

 prove that PK=A'P and PL = AP, and that the normal at 

 P bisects KL. 



510. The foci of an ellipse are the extremities of a diameter 

 of a rectangular hyperbola ; prove that the tangent and normal to 

 the ellipse at any one of the points where it meets the hyperbola 

 are parallel to the asymptotes of the hyperbola. 



511. On a series of parallel chords of a rectangular hyperbola 

 as diameters are described a series of circles ; prove that they will 

 have a common radical axis. 



512. A circle and a rectangular hyperbola intersect in four 

 points, two of which are the extremities of a diameter of the 

 hyperbola ; prove that the other two will be the extremities 

 of a diameter of the circle 1 . 



513. Any chord of a rectangular hyperbola subtends at the 

 extremities of any diameter angles which are either equal or sup- 

 plementary : equal if the extremities of the chord be on the s;ime 

 branch and on the same side of the diameter, or on opposite 

 branches and on opposite sides : otherwise supplementary. 



514. AB is a chord of a circle and a diameter of a rect- 

 angular hyperbola, P any point on the circle ; PA, PB meet the 

 hyperbola again in Q, 7 ; prove that Q, AR will intersect on the 

 circle. 



515. Two points are taken on a rectangular hyperbola and its 

 conjugate, the tangents at which are at right angles to each other; 

 prove that the central radii to the point are also at right angles to 

 each other. 



516. CP, CQ are radii of a rectangular hyperbola, tangents 

 at P, Q meet in jPand intersect CQ, VP respectively in P e J Q' ; 

 prove that a circle can be described about 



