Question

If $|\frac{z}{\left|\overline{z}\right|}-\overline{z}|=1+\left|z\right|$

Then prove that $z$ is a purely imaginary number.

Answer 1

$|z/\left|\overline{z}\right|-\overline{z}|=1+\left|z\right|$
put $z=r(\cos \theta +i\sin \theta )$
$|\cos \theta +i\sin \theta -r(\cos \theta -i\sin \theta )|=1+r$
$\Rightarrow |(1-r)\cos \theta +i(1+r)\sin \theta |=1+r$
$\Rightarrow {(1-r)}^2{\cos }^2\theta +{(1+r)}^2{\sin }^2\theta =1+r^2+2r$
$\Rightarrow 1+r^2-2r\cos 2\theta =1+r^2+2r$
$\Rightarrow \cos 2\theta =\pi$
$\Rightarrow \theta =\frac{\pi }{2}$
Therefore, $z=r(\cos \theta +i\sin \theta )=r(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2})=ir$