Circle of Apollonius in Complex Plane

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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.



Examples

Example: $\cmod {\dfrac {z - 3} {z + 3} } = 2$

The equation:

$\cmod {\dfrac {z - 3} {z + 3} } = 2$

describes a circle embedded in the complex plane whose radius is $4$ and whose center is $\paren {-5, 0}$.


Example: $\cmod {\dfrac {z - 3} {z + 3} } < 2$

The inequality:

$\cmod {\dfrac {z - 3} {z + 3} } < 2$

describes the exterior of a circle embedded in the complex plane whose radius is $4$ and whose center is $\paren {-5, 0}$.


Sources