122 BOOK OF MATHEMATICAL PROBLEMS. 



597. In an ellipse whose axes are in the ratio ^/2 + 1 : 1, a 

 circle whose diameter joins the extremities of two conjugate dia- 

 meters of the ellipse will touch the ellipse. 



598. Two circles have each double contact with an ellipse 

 and touch each other; prove that 



r, r' being the radii ; also that the point of contact of the two 

 circles is equidistant from their two chords of contact with the 

 ellipse. 



599. Two ellipses have common foci S, S', and from a point 

 P on the outer are drawn two tangents PQ, PQf to the inner ; 



. QPQ SPS' . 



prove that cos = : cos 5 is a constant ratio. 



600. The sides of a parallelogram circumscribing an ellipse 

 are parallel to conjugate diameters; prove that the rectangle 

 under the perpendiculars let fall from two opposite angles on any 

 tangent is equal to the rectangle under those from the other two 

 angles. 



601. Prove that the equation 



x y . n r 



- cos a + , sin a cos p / 



l b ^} 



is true at any point of the ellipse 



and hence that the locus of a point, from which if two tangents be 

 drawn to the ellipse the centre of the circle inscribed in the 

 triangle formed by the two tangents and the chord of contact 

 shall lie on the ellipse, is the curve 



" <*'-&' 



