Apollonius and the Hyperbolic Circle

Given a positive number n>1 and two distinct planar focal points A and B, the locus of all points P such that |AP|=n|BP| is a circle of Apollonius of index n Under what conditions is B the center of a hyperbolic circle in the Poincaré disk? In particular, for any circle E in C lying within the unit circle, where the Euclidean center u of E is other than O and its Euclidean radius is ϵ, 0<ϵ<1, there exists another circle D and a point B in E where the center of D is O and its Euclidean radius is δ, 0<δ<1, for which D and E are hyperbolic translates of one another, as are O and B; by rotation symmetry of C about O, we may take A, B, and u as real numbers with v=B, and u+ϵn=A, with 0<u<v<1, and we conclude that nδv=1. That is, B is the hyperbolic center of E as well as the inner Apollonian focus of E whose Apollonian index is n, and A=u+ϵn is the outer focus of E. Furthermore, if P and Q are distinct points on such a circle of Apollonius with index n we have the cross-ratio of the four points (not necessarily collinear) for which |AP||BQ||AQ||BP|=1; with X being an indeterminate point, this identity in turn means that |AP||BX||AQ||BX|=|AP||BQ||AQ||BP| is the equation of a circle, demonstrating why this old Apollonian algorithm forms the core idea for a hyperbolic metric.