Question

If z and ω are two non-zero complex numbers such that $\mid zw\mid =1$ and $arg\left(z\right)-agr\left(w\right)=\frac{\pi }{2}$, then $\overline{z}w$ is

• A. 1
• B. -1
• C. i
• D. -i

Answer 1

The correct option is D -i

$\mid z\mid \mid w\mid =1......\left(i\right)$
and $arg\left(\frac{z}{w}\right)=\frac{\pi }{2}\Rightarrow \frac{z}{w}=i\Rightarrow \mid \frac{z}{w}\mid =1........\left(ii\right)$
From equation (i) and (ii)

$\mid z\mid =\mid w\mid =1and\frac{z}{w}+\frac{\overline{z}}{\overline{}w}=0;z\overline{w}+\overline{z}w=0$

$\overline{z}w=-z\overline{w}=\frac{-z}{w}\overline{w}w;\overline{z}w=-i\mid w{\mid }^2=-i.$