<s>[INST] locus of $z=w - \frac{1}{w}$ If $|w|=2$ , then set of points $z=w -\frac{1}{w}$ is equal to ?
One of my friend helped me like this:
$$|z| = \left| w - \frac{1}{w}\right| \leq |w| + \frac{1}{|w|} = 2 + 0.5 = 2.5 \\ \implies |z| \le 2.5$$
After that I am unable to proceed. Can anybody help me?
 [/INST] Given $w \overline{w} = |w|^2=4\,$, it follows that $z = w - \cfrac{1}{w} = w - \cfrac{\overline{w}}{4}\,$.
Let $z=x+iy$ and $w=u+iv\,$ with $x,y,u,v \in \mathbb{R}\,$, then the above can be written as:
$$
x+iy = u+iv - \frac{1}{4}(u - iv) = \frac{3}{4}u + i\,\frac{5}{4}v \;\;\implies\;\; x = \frac{3}{4}u, \;\;y = \frac{5}{4}v
$$
The condition $|w|^2=4 \iff u^2+v^2=4$ then gives the locus of $z$ as the ellipse:
$$\left(\frac{4}{3}x\right)^2 + \left(\frac{4}{5}y\right)^2 = 4 \;\;\iff\;\; \frac{x^2}{9}+\frac{y^2}{25}=\frac{1}{4}$$
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