BOOK OP MATHEMATICAL PEOBLEMS. 



664. If r,, r g , r a be the focal distances of three points on a 

 conic, and a, /?, y the angles between them, the latus rectum I is 

 given by the equation 



8 sin sin ~ sin 



~ 

 222 sin a sin p sin y 



- - ~ + ' 



the angles a, /?, y being always so taken that their sum is 2ir. 



665. /* is a radius vector from the focus of a conic, Q, Q' two 

 points on it, conjugate with respect to the conic ; prove that the 

 latus rectum 



2SP.SQ 2SP.S& 



666. If PQ, PR be two chords of an ellipse subtending each 

 the same angle at the focus, the tangent at P and the chord QH 

 will intersect on the directrix. 



667. If an ellipse circumscribe a triangle ABC, and the 

 centre of perpendiculars of the triangle be a focus of the ellipse, 

 the latus rectum will be 



47? cos A cos B cos C t 

 A . B . G ' 



R being the radius of the circumscribed circle. 

 668. Prove that any chord of the ellipse 



- = 1 + e cos 

 r 



which is normal at a point where the conic is met by the lines 

 (e + - = sin - (1 - e 2 ) cos Q 



will subtend a right angle at the pole. 



