# Equation of Hyperbola in Complex Plane

## Theorem

Let $\C$ be the complex plane.

Let $H$ be a hyperbola in $\C$ whose major axis is $d \in \R_{>0}$ and whose foci are at $\alpha, \beta \in \C$.

Then $C$ may be written as:

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

where $\cmod {\, \cdot \,}$ denotes complex modulus.

## Proof

By definition of complex modulus:

- $\cmod {z - \alpha}$ is the distance from $z$ to $\alpha$
- $\cmod {z - \beta}$ is the distance from $z$ to $\beta$.

Thus $\cmod {z - \alpha} - \cmod {z - \beta}$ is the difference of the distance from $z$ to $\alpha$ and from $z$ to $\beta$.

This is precisely the equidistance property of the hyperbola.

From Equidistance of Hyperbola equals Transverse Axis, the constant distance $d$ is equal to the transverse axis of $H$.

$\blacksquare$

## Examples

### Example: Foci at $3$ and $-3$, Transverse Axis $4$

The hyperbola in the complex plane whose transverse axis is of length $4$ and whose foci are at the points corresponding to $-3$ and $3$ is given by the equation:

- $\cmod {z + 3} - \cmod {z - 3} = 4$

### Hyperbola Defined by $\map \Im {z^2} = 4$

The equation:

- $\map \Im {z^2} = 4$

describes a hyperbola embedded in the complex plane.

### Hyperbola Defined by $\map \Re {z^2} > 1$

The inequality:

- $\map \Re {z^2} > 1$

describes the area shaded yellow defined by the following hyperbola: