Question

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

• A. $\overline{z}\omega =-i$
• B. $z\overline{\omega }=\frac{-1+i}{\sqrt{2}}$
• C. $\overline{z}\omega =i$
• D. $z\overline{\omega }=\frac{1-i}{\sqrt{2}}$

Answer 1

The correct option is A $\overline{z}\omega =-i$

$\because |zw|=1$
$\Rightarrow |z\overline{w}|=|\overline{z}w|=1$

$\arg z-\arg w=\frac{\pi }{2}$
$\Rightarrow \arg z\overline{w}=\frac{\pi }{2}[\because \arg (\overline{w})=-\arg (w\left)\right]$
$\therefore z\overline{w}=|z\overline{w}|e^{i\left(\arg \right(z\overline{w}\left)\right)}$
$=1\cdot e^{i\pi /2}$
$=i$

$\overline{z\overline{w}}=\overline{z}w=-i$