# Equation of Imaginary Axis in Complex Plane

## Theorem

Let $\C$ be the complex plane.

Let $z \in \C$ be subject to the condition:

$\cmod {z - 1} = \cmod {z + 1}$

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

Then the locus of $z$ is the imaginary axis.

## Proof

 $\displaystyle \cmod {z - 1}$ $=$ $\displaystyle \cmod {z + 1}$ $\displaystyle \leadsto \ \$ $\displaystyle \cmod {z - 1}^2$ $=$ $\displaystyle \cmod {z + 1}^2$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {z - 1} \paren {\overline {z - 1} }$ $=$ $\displaystyle \paren {z + 1} \paren {\overline {z + 1} }$ Modulus in Terms of Conjugate $\displaystyle \leadsto \ \$ $\displaystyle z \overline z - z - \overline z + 1$ $=$ $\displaystyle z \overline z + z + \overline z + 1$ $\displaystyle \leadsto \ \$ $\displaystyle 2 \paren {z + \overline z}$ $=$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle 4 \map \Re z$ $=$ $\displaystyle 0$ Sum of Complex Number with Conjugate $\displaystyle \leadsto \ \$ $\displaystyle \map \Re z$ $=$ $\displaystyle 0$

The result follows by definition of imaginary axis.

$\blacksquare$