Complex Conjugate Coordinates/Examples

Examples of Complex Conjugate Coordinates

Example: $2 x + y = 5$

The equation of the straight line in the plane:

$2 x + y = 5$

can be expressed in complex conjugate coordinates as:

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

Example: $x^2 + y^2 = 36$

The equation of the circle:

$x^2 + y^2 = 36$

can be expressed in complex conjugate coordinates as:

$z \overline z = 36$

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

The equation of the circle:

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

can be expressed in complex conjugate coordinates as:

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

Example: $2 x - 3 y = 5$

The equation of the straight line in the plane:

$2 x - 3 y = 5$

can be expressed in complex conjugate coordinates as:

$\paren {2 i - 3} z + \paren {2 i + 3} \overline z = 10 i$

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

The equation of the ellipse:

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

can be expressed in complex conjugate coordinates as:

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