Find the locus of $w$ $$
\text{Find the locus of $w$, where $z$ is restricted as indicated:} \\
w = z - \frac{1}{z} \\ \text{if } |z| = 2 
$$
I have tried solving this by multiplying both sides by $z$, and then using the quadtratic equation. I get $z = \frac{w \pm \sqrt{w^2+4}}{2}$.
I then set $0 \leq w^2+4 $ But I still have no idea on how to solve this.
Thanks in advance.

let $z = 2(\cos t + i \sin t)$.  Then $$w= u + iv= z - \frac 1 z=2(\cos t + i \sin t ) - \frac 1 2(\cos t - i\sin t).  $$  that gives you $$u = \frac 3 2\cos t, v = \frac 52 \sin t, \, \text{ which is an ellipse   } \frac 49 u^2 + \frac 4 {25} v^2 = 1. $$
