Equation of Line in Complex Plane/Formulation 1
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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.
Proof
From Equation of Straight Line in Plane, the equation for a straight line is:
- $A x + B y + C = 0$
Thus:
\(\ds A x + B y + C\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac A 2 \paren {z + \overline z} + B y + C\) | \(=\) | \(\ds 0\) | Sum of Complex Number with Conjugate | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac A 2 \paren {z + \overline z} + \frac B {2 i} \paren {z - \overline z} + C\) | \(=\) | \(\ds 0\) | Difference of Complex Number with Conjugate | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\frac A 2 + \frac B {2 i} } z + \paren {\frac A 2 - \frac B {2 i} } \overline z + C\) | \(=\) | \(\ds 0\) | gathering terms |
The result follows by setting $\beta := \dfrac A 2 + \dfrac B {2 i}$ and $\gamma := C$.
$\blacksquare$
Sources
- 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: $44$