Complex Conjugate Coordinates/Examples/(x-3)^2 + y^2 = 9

Example of Complex Conjugate Coordinates

The equation of the circle:

$\paren {x - 3}^2 + y^2 = 9$

can be expressed in complex conjugate coordinates as:

$\paren {z - 3} \paren {\overline z - 3} = 9$

Proof

\(\ds \paren {x - 3}^2 + y^2\)

\(=\)

\(\ds 9\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \paren {\paren {x - 3} + i y} \paren {\paren {x - 3} - i y}\)

\(=\)

\(\ds 9\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \paren {\paren {x + i y} - 3} \paren {\paren {x - i y} - 3}\)

\(=\)

\(\ds 9\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \paren {z - 3} \paren {\overline z - 3}\)

\(=\)

\(\ds 9\)

as $z = x + i y$, $\overline z = x - i y$

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Conjugate Coordinates: $117 \ \text{(a)}$