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