140 BOOK OF MATHEMATICAL PROBLEMS. 



G79. The locus of the foot of the perpendicular let fall from 

 a point (X, Y) on any tangent to the ellipse ax 2 +2bxy+cy 9 1 is 



= (ac-b*){x(x-X) + y(y-Y)}, 



the axes being rectangular. Prove that this reduces to a circle 

 and a point-circle, if 



a c o o ac 



680. The equations determining the foci of the conic 



ax* + 2bxy + cy* = d 

 y (x + y cos o>) _ x (y + x cos CD) ^ d 



a cos <u b c cos o> 6 ~~ ac b* ' 



to being the angle between the axes. 



681. The general equation of a conic confocal with the 

 conic ax* + 2bxy + cy*=I is 



(x* + y*) (ac b*) + \ (ax* + 2bxy + cy*) = - j 



X 



and the given conic is cut at right angles by any conic whose 

 equation is 



X (c a) 2ac 

 (X-a)x 9 + ^ -xy-(\ + c)y* = l. 



682. The equation of the equi-conjugates of the conic 



ax* + 2bxy + cy*=l 

 ax* + 2bxy + cy*_2 (x* + y*) 



~y~A 



ac-b* a + c 



683. The equation of the equi-conjugates of the conic 



ax* + 2bxy + cy*=l 

 is (a - c) (ax* - cy*) + 2x (bx + cy) (b-a cos w) 



+ 2y (ax + by) (b - c cos o>) = ; 

 when <o is the angle between the axes. 



