Complex Conjugate Coordinates/Examples/2x - 3y = 5
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Example of Complex Conjugate Coordinates
The equation of the straight line in the plane:
- $2 x - 3 y = 5$
can be expressed in complex conjugate coordinates as:
- $\paren {2 i - 3} z + \paren {2 i + 3} \overline z = 10 i$
Proof
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:
\(\ds 2 x - 3 y\) | \(=\) | \(\ds 5\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \paren {\frac {z + \overline z} 2} - 3 \paren {\frac {z - \overline z} {2 i} }\) | \(=\) | \(\ds 5\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {z + \overline z} 2 i - 3 \paren {z - \overline z}\) | \(=\) | \(\ds 10 i\) | |||||||||||
\(\ds \leadsto= \ \ \) | \(\ds 2 i z + 2 i \overline z - 3 z + 3 \overline z\) | \(=\) | \(\ds 10 i\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {2 i - 3} z + \paren {2 i + 3} \overline z\) | \(=\) | \(\ds 10 i\) |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Conjugate Coordinates: $117 \ \text{(b)}$