Complex Conjugate Coordinates/Examples/x^2 + y^2 = 36/Proof 1

Example of Complex Conjugate Coordinates

The equation of the circle:

$x^2 + y^2 = 36$

can be expressed in complex conjugate coordinates as:

$z \overline z = 36$

Proof

\(\ds x^2 + y^2\)

\(=\)

\(\ds 36\)

\(\ds \leadstoandfrom \ \ \)

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

\(=\)

\(\ds 36\)

\(\ds \leadstoandfrom \ \ \)

\(\ds z \overline z\)

\(=\)

\(\ds 36\)

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

$\blacksquare$

Sources

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