Equation of Ellipse in Complex Plane/Examples/Foci at (0, 2), (0, -2), Major Axis 10
Jump to navigation
Jump to search
Example of Equation of Ellipse in Complex Plane
The ellipse in the complex plane whose major axis is of length $10$ and whose foci are at the points corresponding to $\tuple {0, 2}$ and $\tuple {0, -2}$ is given by the equation:
- $\cmod {z + 2 i} + \cmod {z - 2 i} = 10$
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 points $\tuple {0, 2}$ and $\tuple {0, -2}$ correspond to the imaginary numbers $2 i$ and $-2 i$ respectively.
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: $72 \ \text {(b)}$