Question

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

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

Answer 1

Answer: (B)

$z\overline{w}=i$

$\left|z\right|.\left|w\right|=1,z=r{e}^{i(\theta +\pi /2)};w=\frac{1}{r}{e}^{i\theta }$
$\overline{z}w=e^{-i(\theta +\pi /2)}.e^{i\theta }=e^{-\left(\pi /2\right)}=-i$
$z\overline{w}=e^{i(\theta +\pi /2)}.e^{i\theta }=e^{\left(\pi /2\right)}=i$