Equation of Ellipse in Complex Plane/Examples/Foci at (0, 2), (0, -2), Major Axis 10

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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.

$\blacksquare$


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