1 ['2 BOOK OF MATHEMATICAL PROBLEMS. 



691. Trace the locus of the equation 



+*.=. 



x+y-a x + y-b 

 for values of m > and < 1. 

 C92. Trace the conies 



0, 



shewing that they touch each other two and two. 



C93. Two parabolas are so situated that a circle can be de- 

 scribed through their four points of intersection ; prove that the 

 distance of the centre of this circle from the axis of either para- 

 bola is equal to half the latus rectum of the other. 



C94. A hyperbola is drawn touching the axes of an ellipse 

 and the asymptotes of the hyperbola touch the ellipse; prove that 

 the centre of the hyperbola lies on one of the equi-conjugates of 

 the ellipse. 



695. Five fixed points are taken, no three of which are in 

 one straight line, and five conies are described, each bisecting all 

 the lines joining four of the points, two and two: prove that these 

 conies will have one common point. 



696. A, P and B, Q are points taken respectively on two 

 parallel fixed straight lines, A, B being fixed points and P, Q 

 variable points, subject to the condition that AP . BQ is constant; 

 prove that PQ touches a fixed conic which will be an ellipse or 

 hyperbola according as P, Q are on the same or opposite sides of 

 AB. 



C97. One side AB of a rectangle ABCD slides between two 

 rectangular axes ; prove that the loci of (7, D are ellipses whose 

 area is independent of the magnitude of AB ; and that the angle 



2/?(7 

 between their axes is cot' 1 -~- . If in any position AB make 



an angle with the axis of y, and a, ft be the angles which the 



