If P moves such that PA/PB = k (constant, k != 1), where A and B are fixed, the locus of P is a circle called the Apollonius circle.

Let A=(a,0), B=(-a,0), PA/PB=k. Then (h-a)^2+$k^2$ = $k^2$*((h+a)^2+$k^2$). Expanding and simplifying gives a circle equation.

Special case: k=1 gives PA=PB, which is the perpendicular bisector of AB (a line, not a circle). This is the degenerate case.

The Apollonius circle divides the segment AB internally and externally in the ratio k:1. Its center lies on the line AB.