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:
- $\cmod {z - \alpha} + \cmod {z - \beta} = d$
The rest of the result follows from Equation of Ellipse in Reduced Form: Cartesian Frame.
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $13 \ \text{(b)}$
- 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: $74$