Question

If $z$ and $\omega$ are two non-zero complex numbers such that $\left|z\omega \right|=1$ and $\arg z-\arg \omega =\frac{\pi }{2}$, then
$\overline{z}\omega =$

• A. $1$
• B. $-1$
• C. $i$
• D. $-i$

Answer 1

The correct option is D $-i$

$\left|z\omega \right|=1\Rightarrow \left|z\right|\left|\omega \right|=1$
$\therefore \left|\omega \right|=\frac{1}{\left|z\right|}$ .....$\left(1\right)$
Let $z=r{e}^{i\theta }$ $\therefore \overline{z}=r{e}^{-i\theta }$
Given, $Arg\omega =Argz-\pi /2=\theta -\pi /2$
$\therefore \omega =\frac{1}{r}{e}^{i(\theta -\pi /2)}$ by $\left(1\right)$
$\overline{z}\omega =\left({re}^{-i\theta }\right).\omega =\frac{1}{r}{e}^{i(\theta -\pi /2)}=\omega =\frac{1}{r}{e}^{-i\pi /2}$
$=\cos \frac{\pi }{2}-i\sin \frac{\pi }{2}=-i$