Question

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

• A. 1
• B. i
• C. -1
• D. -i

Answer 1

The correct option is D -i

$|z||\omega |=1Asarg\left(\frac{z}{\omega }\right)=\frac{\pi }{2}.\therefore \frac{z}{\omega }=ki\therefore \left|\begin{array}{c}\frac{z}{\omega }=k\end{array}\right|from\left(i\right)\&\left(ii\right),|z|=\sqrt{k},|\omega |=\frac{1}{\sqrt{k}}\therefore \frac{z}{\omega }+\frac{\overline{z}}{\overline{\omega }}=0\Rightarrow z\overline{\omega }+\overline{z}\omega =0\Rightarrow \overline{z}\omega =-z\overline{\omega }=\frac{-z}{\omega }\times \overline{\omega }\times \omega \Rightarrow \overline{z}\omega =-ki.\frac{1}{k}=-i$