OIMU 2005 Problem 4

A variable tangent $t$ to the circle $C_1$ , of radius $r_1$ , intersects the circle $C_2$ , of radius $r_2$ in $A$ and $B$ . The tangents to $C_2$ through $A$ and $B$ intersect in $P$ . Find, as a function of $r_1$ and $r_2$ , the distance between the centers of $C_1$ and $C_2$ such that the locus of $P$ when $t$ varies is contained in an equilateral hyperbola.Note: A hyperbola is said to be equilateral if its asymptotes are perpendicular.