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 $H$.