Equation of Circle in Complex Plane/Examples/Straight Line y = -3
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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
\(\ds \overline z\) | \(=\) | \(\ds z + 6 i\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds z - \overline z\) | \(=\) | \(\ds -6 i\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds i z - i \overline z\) | \(=\) | \(\ds -6 i^2\) | multiplying both sides by $i$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds i z - \overline i \overline z\) | \(=\) | \(\ds 6\) |
$\blacksquare$
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:
\(\ds i z - \overline i \overline z\) | \(=\) | \(\ds 6\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds i \paren {x + i y} - i \paren {x - i y}\) | \(=\) | \(\ds 6\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds i x + i^2 y - i x + i^2 y\) | \(=\) | \(\ds 6\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 y\) | \(=\) | \(\ds -6\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds -3\) |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Conjugate Coordinates: $116 \ \text{(d)}$