Locus in Complex plane

Question

Could someone help me out with this one

Show that the locus of w as z varies with |z| = 1, where w is given by $$w^2=\frac {1-z}{1+z}$$ is a pair of straight lines.

Answer 1

This should help to get you started: let $z = e^{i \theta} = \cos \theta + i \sin \theta$ where $\theta \in [0,2\pi)$. We have $$ \begin{align} w^2 &= \frac{1 - z}{1 + z} = \frac{1 - e^{i\theta}}{1 + e^{i \theta}}\cdot \frac{1 + e^{-i\theta}}{1 + e^{-i\theta}}\\ &= \frac{-e^{i\theta} + e^{-i\theta}}{2 + e^{i \theta} + e^{-i \theta}} = -i\frac{\sin \theta}{1+\cos \theta} \end{align} $$ Since $w^2$ is purely imaginary, it follows that $|\operatorname{Re}(w)| = |\operatorname{Im}(w)|$. Now, show that this locus is the set of all such points.