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:
