Equation of Circle in Complex Plane/Formulation 2/Proof 1

Proof
By setting:
 * $\beta := -\alpha a$ and $\gamma = \alpha a \overline a - r^2$

we have:
 * $\alpha z \overline z + \beta z + \overline \beta \overline z + \gamma = 0$

We have that:

If $\alpha = 0$ and $\beta \ne 0$ the equation devolves to:


 * $\beta z + \overline \beta \overline z + \gamma = 0$

which from Equation of Line in Complex Plane: Formulation 1 is the equation of a straight line.

The result follows.