130 BOOK OF MATHEMATICAL PROBLEMS. 



will be capable of having triangles circumscribing the first and 

 inscribed in the second, if 



641. Two straight lines are drawn parallel to the major axis 

 at a distance - from it ; prove that the part of any tangent i 



cepted between them will be divided by the point of contact into 

 two parts subtending equal angles at the centre. 



642. A circle has its centre in the major axis of an ellipse, 

 and triangles can be inscribed in the circle whose sides touch the 

 ellipse j prove that the circle must touch the two circles 



x" + (y by = a s . 



643. Two tangents are drawn to an ellipse from a point 

 (X, Y) ; prove that the rectangle under the perpendiculars from 

 any point of the ellipse on the tangents bears to the square on the 

 perpendicular to the chord of contact the ratio 1 : X; where 



644. If a triangle be inscribed in an ellipse, and the centre of 

 perpendiculars of the triangle be one of the foci, the sides of the 

 triangle will touch one of the circles 



IY. Hyperbola, referred to its axes, or asymptotes. 

 645. Prove that the four equations 



represent respectively the portions of the hyperbola 



6V-y = a 2 6 2 

 which lie in the four quadrants. 



