Equation of Ellipse in Complex Plane/Examples/Foci at -3, 3, 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 $-3$ and $3$ is given by the equation:

$\cmod {z + 3} + \cmod {z - 3} = 10$

and also as:

$\dfrac {x^2} {25} + \dfrac {y^2} {16} = 1$


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:

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


Equation of Ellipse in Complex Plane-Examples-Foci at -3, 3, Major Axis 10.png


The rest of the result follows from Equation of Ellipse in Reduced Form: Cartesian Frame.

$\blacksquare$


Sources