Equation of Hyperbola in Complex Plane/Examples/Foci at 3, -3, Transverse Axis 4

Example of Equation of Hyperbola in Complex Plane

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:

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

Proof

From Equation of Hyperbola in Complex Plane, the hyperbola whose transverse 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 result follows.

$\blacksquare$

Sources

• 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: $71 \ \text {(c)}$