Equation of Ellipse in Complex Plane/Examples/Foci at -2i, 2i, Major Axis 6

Example of Equation of Ellipse in Complex Plane

The ellipse in the complex plane whose major axis is of length $6$ and whose foci are at the points corresponding to $-2 i$ and $2 i$ is given by the equation:

$\cmod {z + 2 i} + \cmod {z - 2 i} = 6$

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

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $71 \ \text {(b)}$