Equation of Ellipse in Complex Plane/Examples
Examples of Equation of Ellipse in Complex Plane
Example: Foci at $1$ and $i$, Major Axis $4$
The ellipse in the complex plane whose major axis is of length $4$ and whose foci are at the points corresponding to $1$ and $i$ is given by the equation:
- $\cmod {z - 1} + \cmod {z - i} = 4$
Example: Foci at $-3$ and $3$, Major Axis $10$
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:
- $\dfrac {x^2} {25} + \dfrac {y^2} {16} = 1$
Example: Foci at $-2 i$ and $2 i$, Major Axis $6$
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:
- $\cmod {z + 2 i} + \cmod {z - 2 i} = 6$
Example: Foci at $-2 i$ and $2 i$, Major Axis $6$
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:
- $\cmod {z + 2 i} + \cmod {z - 2 i} = 6$
Example: Foci at $\tuple {2, -3}$ and $\tuple {-2, 3}$, Major Axis $10$
The inequality:
- $\cmod {z + 2 - 3 i} + \cmod {z - 2 + 3 i} < 10$
defines the inside of the ellipse in the complex plane whose major axis is of length $10$ and whose foci are at the points corresponding to $\tuple {2, -3}$ and $\tuple {2, -3}$.