Question: If $z$ and $\omega$ are two complex numbers such that $|z \omega|=1$ and $\arg (z)-\arg (\omega)=\frac{\pi}{2}$, then:
(1) $\bar{z} \omega=i$
(2) $z \bar{\omega}=\frac{-1+i}{\sqrt{2}}$
(3) $\bar{z} \omega=-i$
(4) $z \bar{\omega}=\frac{1-i}{\sqrt{2}}$
Correct Option: , 3
Solution:
Given $|z \omega|=1$
and $\arg \left(\frac{z}{\omega}\right)=\frac{\pi}{2}$
