Equation of Ellipse in Complex Plane/Examples/Foci at 1, i, Major Axis 4

Example of Equation of Ellipse in Complex Plane

The ellipse in the complex plane whose major axis is of length $4$ and whose foci are at the points corresponding to $1$ and $i$ is given by the equation:

$\cmod {z - 1} + \cmod {z - i} = 4$

Proof

From Equation of Ellipse in Complex Plane, the ellipse whose major axis is $d$ and whose foci are at the points corresponding to $\alpha$ and $\beta$ is given by:

$\cmod {z - \alpha} + \cmod {z - \beta} = d$

The result follows.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: Exercise $3$.