Complex Conjugate Coordinates/Examples/4x^2 + 16y^2 = 25

Example of Complex Conjugate Coordinates

The equation of the ellipse:

$4 x^2 + 16 y^2 = 25$

can be expressed in complex conjugate coordinates as:

$3 \paren {z^2 + \overline z^2} - 10 z \overline z + 25 = 0$

Proof

This can be written in the form:

$a \paren {x^2 + y^2} + 2 b \paren {x^2 - y^2} = 25$

where:

\(\ds a + 2 b\)

\(=\)

\(\ds 4\)

\(\ds a - 2 b\)

\(=\)

\(\ds 16\)

\(\ds \leadsto \ \ \)

\(\ds 2 a\)

\(=\)

\(\ds 20\)

\(\ds \leadsto \ \ \)

\(\ds a\)

\(=\)

\(\ds 10\)

\(\ds \leadsto \ \ \)

\(\ds 2 b\)

\(=\)

\(\ds -6\)

\(\ds \leadsto \ \ \)

\(\ds b\)

\(=\)

\(\ds -3\)

This is because:

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

\(=\)

\(\ds z \overline z\)

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

\(=\)

\(\ds z^2 + \overline z^2\)

So:

\(\ds 4 x^2 + 16 y^2\)

\(=\)

\(\ds 25\)

\(\ds \leadsto \ \ \)

\(\ds 10 \paren {x^2 + y^2} + 2 \paren {-3} \paren {x^2 - y^2}\)

\(=\)

\(\ds 25\)

\(\ds \leadsto \ \ \)

\(\ds 10 z \overline z - 3 \paren {z^2 + \overline z^2}\)

\(=\)

\(\ds 25\)

\(\ds \leadsto \ \ \)

\(\ds 3 \paren {z^2 + \overline z^2} - 10 z \overline z + 25\)

\(=\)

\(\ds 0\)

$\blacksquare$

Sources

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