Complex Conjugate Coordinates/Examples/2x + y = 5

Example of Complex Conjugate Coordinates

The equation of the straight line in the plane:

$2 x + y = 5$

can be expressed in complex conjugate coordinates as:

$\paren {2 i + 1} z + \paren {2 i - 1} \overline z = 10 i$

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:

\(\ds 2 x + y\)

\(=\)

\(\ds 5\)

\(\ds \leadsto \ \ \)

\(\ds 2 \paren {\frac {z + \overline z} 2} + \paren {\frac {z - \overline z} {2 i} }\)

\(=\)

\(\ds 5\)

\(\ds \leadsto \ \ \)

\(\ds \paren {z + \overline z} 2 i + \paren {z - \overline z}\)

\(=\)

\(\ds 10 i\)

\(\ds \leadsto \ \ \)

\(\ds 2 i z + 2 i \overline z + z - \overline z\)

\(=\)

\(\ds 10 i\)

\(\ds \leadsto \ \ \)

\(\ds \paren {2 i + 1} z + \paren {2 i - 1} \overline z\)

\(=\)

\(\ds 10 i\)

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Complex Conjugate Coordinates: $43 \ \text {(a)}$