Circle of Apollonius in Complex Plane

Theorem
Let $\C$ be the complex plane.

Let $\lambda \in \R$ be a real number such that $\lambda \ne 0$ and $\lambda \ne 1$.

Let $a, b \in \C$ such that $a \ne b$.

The equation:
 * $\cmod {\dfrac {z - a} {z - b} } = \lambda$

decribes a circle of Apollonius $C$ in $\C$ such that:


 * if $\lambda < 0$, then $a$ is inside $C$ and $b$ is outside
 * if $\lambda > 0$, then $b$ is inside $C$ and $a$ is outside.

If $\lambda = 1$ then $z$ describes the perpendicular bisector of the line segment joining $a$ to $b$.

Proof
By the geometry, the locus described by this equation is a circle of Apollonius.