Equation of Circle in Complex Plane/Examples/Straight Line y = -3

Example of Use of Equation of Circle in Complex Plane
The equation:
 * $\overline z = z + 6 i$

describes the straight line $y = -3$ embedded in the complex plane

Proof
This is an instance of the Equation of Circle in Complex Plane: Formulation 2:
 * $\alpha z \overline z + \beta z + \overline \beta \overline z + \gamma = 0$

where $\alpha = 0$.

This is a straight line $\alpha = 0$ and $\beta \ne 0$.

Hence: