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:

This is because:

So: