Complex Conjugate Coordinates/Examples/x^2 + y^2 = 36
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Example of Complex Conjugate Coordinates
- $x^2 + y^2 = 36$
can be expressed in complex conjugate coordinates as:
- $z \overline z = 36$
Proof 1
| \(\ds x^2 + y^2\) | \(=\) | \(\ds 36\) | ||||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \paren {x + i y} \paren {x - i y}\) | \(=\) | \(\ds 36\) | |||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds z \overline z\) | \(=\) | \(\ds 36\) | as $z = x + i y$, $\overline z = x - i y$ |
$\blacksquare$
Proof 2
We have that:
| \(\ds z\) | \(=\) | \(\ds x + i y\) | ||||||||||||
| \(\ds \overline z\) | \(=\) | \(\ds x - i y\) |
and so:
| \(\ds x\) | \(=\) | \(\ds \frac {z + \overline z} 2\) | ||||||||||||
| \(\ds y\) | \(=\) | \(\ds \frac {z - \overline z} {2 i}\) |
Hence:
| \(\ds \paren {\frac {z + \overline z} 2}^2 + \paren {\frac {z - \overline z} {2 i} }^2\) | \(=\) | \(\ds 36\) | ||||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \paren {\frac {z^2 + 2 z \overline z + \overline z^2} 4} + \paren {\frac {z^2 - 2 z \overline z + \overline z^2} {4 i^2} }\) | \(=\) | \(\ds 36\) | |||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \frac {z^2 - z^2 + 2 z \overline z + 2 z \overline z + \overline z^2 - \overline z^2} 4\) | \(=\) | \(\ds 36\) | |||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds z \overline z\) | \(=\) | \(\ds 36\) |
$\blacksquare$