Finding the Locus if $z$ Purely Imaginary/Real

Question

If $\omega=\frac{z-1}{z+i}$, find the locus if $\omega$ is:

1. Purely Imaginary
2. Purely real

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

One of my friends suggested that I rationalise $\omega$ and split it into its real and imaginary parts like below:

$\omega=\frac{z^2+z}{z^2+1}-i\frac{z+1}{z^2+1}$

but I'm not convinced that this works, because $z$ could stand for a complex number as well? And wouldn't that make this whole process invalid?

Answer 1

Hint:

$z=x+yi$, $z\not=-i$

i. $\omega = ai$ with $a\in \mathbf{R}$

ii. $\omega = a$ with $a\in \mathbf{R}$