Question

If z and $\omega$are two non - zero complex numbers such that $\left|z\omega \right|$=1 and Argz - 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

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