Hi again, just a quick question about a transformation in the complex plane.

Here's the question:

The transformation $\displaystyle T$ from the complex z plane to the complex w plane is given by $\displaystyle w = \frac{z+1}{z+i}$.

Show that $\displaystyle T$ maps points on the half line $\displaystyle arg{z} = \frac{\pi}{4}$ into points on the circle $\displaystyle |w| = 1$ in the w plane.

The way I did this is by $\displaystyle |\frac{z+1}{z+i}| = 1$, leading to $\displaystyle |z+1| = |z+i|$.

Now if we map the loci of these points we get the half line $\displaystyle arg{z} = \frac{\pi}{4}$.

Just a few questions, firstly is this right? Is it ok to do it this way, instead of starting with the $\displaystyle arg{z} = \frac{\pi}{4}$, and is there another way to do this?

Thanks in advance for the help