Question

If z and w are two nonzero complex numbers such that $\left|zw\right|=1$ and arg $z-$arg $w=\frac{\pi }{2}$ then $\overline{z}w$ is equal to

• A. $-1$
• B. $i$
• C. $-i$
• D. $1$

Answer 1

The correct option is C $-i$

$\left|zw\right|=1\&argz-argw=\frac{\pi }{2}$
Let $\left|z\right|=r\Rightarrow \left|w\right|=\frac{1}{r}$
let $z=\left|z\right|(\cos A+i\sin A)=rcisA\&w=\left|w\right|(\cos B+i\sin B)=\frac{1}{r}cisB$
$\because argz-argw=\frac{\pi }{2}$
$\Rightarrow A-B=\frac{\pi }{2}$ ...(1)
$\overline{z}w=\left(rcis(-A)\right)\left(\frac{1}{r}cisB\right)=cis(-(A-B))$
$\Rightarrow \overline{z}w=cis(-\frac{\pi }{2})=-i$ ...{ From 1 }
Hence, option 'C' is correct.