Equation of Line in Complex Plane/Formulation 1

Theorem

Let $\C$ be the complex plane.

Let $L$ be a straight line in $\C$.

Then $L$ may be written as:

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

where $\gamma$ is real and $\beta$ may be complex.

$A x + B y + C = 0$

Thus:

$\ds A x + B y + C$

$=$

$\ds 0$

$\ds \leadsto \ \ $

$\ds \frac A 2 \paren {z + \overline z} + B y + C$

$=$

$\ds 0$

$\ds \leadsto \ \ $

$\ds \frac A 2 \paren {z + \overline z} + \frac B {2 i} \paren {z - \overline z} + C$

$=$

$\ds 0$

$\ds \leadsto \ \ $

$\ds \paren {\frac A 2 + \frac B {2 i} } z + \paren {\frac A 2 - \frac B {2 i} } \overline z + C$

$=$

$\ds 0$

gathering terms

The result follows by setting $\beta := \dfrac A 2 + \dfrac B {2 i}$ and $\gamma := C$.

$\blacksquare$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: Exercise $5$.
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Complex Conjugate Coordinates: $44$