Let $\omega_1$ and $\omega_2$ be circles with respective centers $O_1$ and $O_2$ and respective radii $r_1$ and $r_2$, and let $k$ be a real number not equal to 1. Prove that the set of points $P$ such that $$ PO_1^2 - r_1^2 = k(PO_2^2 - r_2^2) $$ is a circle.
We know that when $r_1=0,r_2=0$, the problem reduces to the definition of Apollonius' circle. But how to show the same fact for such a case with pure geometry method ?