Mathematics - Complex Number Question with Solution | TestHub

$\begin{array}{rl}& |z|=1\\ & \left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1\\ & \Rightarrow \left|z^2+(\bar{z}{)}^2\right|=1\end{array}$
Let $z=x+iy$
$\begin{array}{rl}& \Rightarrow \left|(x+iy{)}^2+(x-iy{)}^2\right|=1\\ & \Rightarrow \left|2x^2-2y^2\right|=1\\ & \Rightarrow \left|x^2-y^2\right|=\frac{1}{2}\\ & \Rightarrow x^2-y^2=\frac{\pm 1}{2}\\ & \text{ and }{x}^2+y^2=1\end{array}$
Case I: $x^2-y^2=\frac{1}{2}$
Case II: $x^2-y^2=-\frac{1}{2}$