CONIC SECTIONS, ANALYTICAL. 1 1 "> 



584. A parabola touches the sides of a triangle ABC in 

 A', B y C", and is a point such that OA' bisects the angle BOO, 

 and OR, 00' bisect the external angles between 00, OA, and OA, 

 OB, respectively ; prove that OA=OB + 00. 



585. A triangle circumscribes the circle se* + y* = a*, and two 

 angular points lie on the circle (a - 2a) f + ^ = 20*; prove that the 

 third angular point lies on the parabola 



3a 



Prove also that the curves have two real common tangents. 



586. Two parabolas have a common focus, axes inclined at an 

 angle 2o, and are such that triangles can be inscribed in one 

 whose sides touch the other; prove that J f = 4J 1 cos t a, I i9 l t 

 being their latent recta. 



587. A circle is described with its centre at a point P of a 

 parabola, and its radius equal to twice the normal at P; prove 

 that triangles can be inscribed in the parabola whose aides touch 

 the circle. 



588. Two parabolas A, B have their axes parallel, and the 

 latus rectum of B is four times that of A ; prove that an infinite 

 number of triangles can be inscribed in A whose sides are tangents 

 to B. 



III. Ellipse referred to iU axes. 



The equation of the ellipse in the following questions is always 

 supposed to be -| + W- 1, unless otherwise stated The point 



whose eccentric angle is 0, is called the point 0. The oooen- 

 v is denoted by e. The tangent and normal at the point 

 are respectively 



8 J 



