98 BOOK OF MATHEMATICAL PROBLEMS. 



two sides of the parallelogram ; prove that the rectangle under 

 the segments o these lines made by the parallelogram are in 

 a constant ratio. 



496. Any two central conies in the same plane have two con- 

 jugate diameters of the one parallel respectively to two conjugate 

 diameters of the other ; and in general no more. 



497. In two similar and similarly situated ellipses are taken 

 two parallel chords PP', QQ ; PQ, PQ meet the two conies 

 in R, S-, R f , S' respectively; prove that RR', SS' are parallel to 

 each other. Also QQ', RR', and PP', SS' intersect in points lying 

 on a fixed straight line. 



498. A circle is described touching the focal distances of any 

 point on a given conic, and passing through a given point on the 

 major axis; prove that it will meet the major axis in another 

 fixed point. The given point must be between the foci for a 

 hyperbola, and beyond them for an ellipse. 



499. A circle described on the part of the tangent at P 

 intercepted between the tangents at the ends of the major axis 

 meets the conic again in Q ; prove that the ordinate of Q is to 

 the ordinate of P as the minor axis to the sum of the minor axis 

 and the diameter conjugate to P. 



500. If a conic be inscribed in a triangle ABC 'and have 

 its focus at 0; and if the angles BOG, GO A, AOB be denoted 

 by A', B f , C', 



OA sin A OB sin B OC sin G 



sin (A - A ) sin (B f - B) sin (G' - C) 



With what convention will this be true if be a point without 

 the triangle 1 



501. OA, OB are tangents to a conic, a straight line is drawn 

 meeting OA, OB in Q, Q', AB in R, and the conic in P, f* 

 prove that QP.PQ' : QF.FQ' :: RP a : RP" ', and that, fora 



