complex numbers - What does $lvert z-a rvert = mathit Re(z)+a$ look like?

What does a loci with the equation look like?

$\lvert z-a \rvert = \mathit Re(z)+a$

This is for the applying complex numbers topic of an advanced HSC maths course. I was asked to describe the loci.

I know that $\lvert z-a \rvert$ would get me either a perpendicular bisector or a circle. I also know that $\mathit Re(z)$ refers to the horizontal values on the complex plane. But I just can't imagine what it looks like.

Answer

We can treat complex numbers $z = x + iy$ as equations over $(x, y) \in \mathbb R^2$, and use a geometry plotter to plot them. In this case, the equation system is:

$$\begin{align*} |x + iy - a| &= Re(x+iy) + a \\ |(x - a) + iy| &= x + a\\ \sqrt{(x-a)^2 + y^2} &= x + a \\ (x-a)^2 + y^2 &= (x + a)^2 \end{align*}$$

One can use a tool like Desmos for plotting curves like these. In this case, here is a playable version of the graph with $a$ as a parameter.

The image for one choice of $a$ is: