Equation of Circle in Complex Plane/Formulation 2

Theorem
Let $\C$ be the complex plane.

Let $C$ be a circle in $\C$.

Then $C$ may be written as:
 * $\alpha z \overline z + \beta z + \overline \beta \overline z + \gamma = 0$

where:
 * $\alpha \in \R_{\ne 0}$ is real and non-zero
 * $\gamma \in \R$ is real
 * $\beta \in \C$ is complex such that $\cmod \beta^2 > \alpha \gamma$.

The curve $C$ is a straight line $\alpha = 0$ and $\beta \ne 0$.