CONIC SECTIONS, ANALYTICAL. 141 



684. The tangents to the two conies 



, , 



a b a b a + b 



at any common point are at right angles. If both curves be 

 hyperbolas, they will have four real common tangents. 



685. If a common tangent be drawn to the Conies 



* tf i ^'-LQX^ y 1 , n i \*\*+ v n- 



. + it = 1, i + *A = 7i + (1 + A ) -= - = U : 



a* 6' a& 6 3 'a 6 



it will subtend a right angle at the centre. 



686. Prove that in general two parabolas can be descriU.l 

 passing through the points of intersection of the conies 



ca? + by 8 + c + 2ay + 2b'x + 2c'xy = 0, 

 2 C'xy = ; 



iiat the axes of these parabolas will be at right angles to each 

 other if 



a-b~ A-3' 

 687. The equation of the director circle of the conic 



f(x, y)= 

 is x* 



688. The equation of the asymptotes is 



(G" - ab)f(x, y) = aa"+bb n - 2a'b'c. 



689. The equation of the ch>rd whirh passes tlirou-1 

 (X t Y) and is bisected at that ]> >int is 



(x-JT)(a-V+cT + 6') + (y- } >,r+a) = 0; 



and the equation of the tangents drawn through the ori- 

 (ae - b^ x 9 + 2 (<- a'V) xy -f (be - a") y> - 0. 



690. The foci are given by the equations 



x (aa' - 6V) + y (W - c'a') + a'6' - d = ay (* - ao), 

 - c V) - 2y (rut' - 6V) + f - ca - a" + be - (x 9 - /) (c" - ab). 



