On the plane, two points $A$ and $B$ are given. Find the locus of points $M$ such that $A M: B M=k$ (Apollonian circle).

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When $k=1$, we obtain the perpendicular bisector of the segment $A B$. In the following, we will assume that $k \neq 1$.

We introduce a coordinate system on the plane such that points $A$ and $B$ have coordinates $(-a, 0)$ and $(a, 0)$, respectively. If point $M$ has coordinates $(x, y)$, then $\frac{A M^{2}}{B M^{2}}=\frac{(x+a)^{2}+y^{2}}{(x-a)^{2}+y^{2}}$. The equation $\frac{A M^{2}}{B M^{2}}=k^{2}$ can be transformed into

$$
\left(x+\frac{1+k^{2}}{1-k^{2}} a\right)^{2}+y^{2}=\left(\frac{2 k a}{1-k^{2}}\right)^{2}
$$

This equation represents a circle with center $\left(-\frac{1+k^{2}}{1-k^{2}} a, 0\right)$ and radius $\frac{2 k a}{\left|1-k^{2}\right|}$.