Question:
If $\mathrm{z}$ and $\mathrm{w}$ are two complex numbers such that
$|z w|=1$ and $\arg (z)-\arg (w)=\frac{\pi}{2}$, then :
Correct Option: , 2
Solution:
$|\mathrm{z}| .|\mathrm{w}|=1 \quad \mathrm{z}=\mathrm{re}^{\mathrm{i}(\theta+\pi / 2)}$ and $\mathrm{w}=\frac{1}{\mathrm{r}} \mathrm{e}^{\mathrm{i} \theta}$
$\bar{z} . w=e^{-i(\theta+\pi / 2)} \cdot e^{i \theta}=e^{-i(\pi / 2)}=-i$
$z \cdot \bar{w}=e^{i(\theta+\pi / 2)} \cdot e^{-i \theta}=e^{i(\pi / 2)}=i$