Equation of Circle in Complex Plane/Examples/Straight Line x = 2
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Example of Use of Equation of Circle in Complex Plane
The equation:
- $z + \overline z = 4$
describes the straight line $x = 2$ 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 if and only if $\alpha = 0$ and $\beta \ne 0$.
Hence:
\(\ds z + \overline z\) | \(=\) | \(\ds 4\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {x + i y} + \paren {x - i y}\) | \(=\) | \(\ds 4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 x\) | \(=\) | \(\ds 4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds 2\) |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Conjugate Coordinates: $116 \ \text{(c)}$