If z and w are two complex numbers such that
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 :

1. $\overline{Z W}=i$

2. $\bar{Z} W=-1$

3. $z \bar{w}=\frac{1-i}{\sqrt{2}}$

4. $z \bar{w}=\frac{-1+i}{\sqrt{2}}$

Correct Option: , 2

Solution:

$|\mathrm{z}| .|\mathrm{w}|=1 \quad \mathrm{z}=\mathrm{re}^{\mathrm{i}(\theta+\pi / 2)}$ and $\mathrm{w}=\frac{1}{\mathrm{r}} \mathrm{e}^{\mathrm{i} \theta}$

$\bar{z} . w=e^{-i(\theta+\pi / 2)} \cdot e^{i \theta}=e^{-i(\pi / 2)}=-i$

$z \cdot \bar{w}=e^{i(\theta+\pi / 2)} \cdot e^{-i \theta}=e^{i(\pi / 2)}=i$