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If $z$ and $\omega$ are two complex numbers such that $|z \omega|=1$ and $\arg (z)-\arg (\omega)=\frac{3 \pi}{2}$, then $\arg \left(\frac{1-2 \bar{z} \omega}{1+3 \bar{z} \omega}\right)$ is:
(Here arg(z) denotes the principal argument of complex number $z$ )
$\frac{3 \pi}{4}$
$-\frac{\pi}{4}$
$-\frac{3 \pi}{4}$
$\frac{\pi}{4}$
Solution
As $|z \omega|=1$
$\Rightarrow|z|=r$, then $|\omega|=\frac{1}{r}$
Let $\arg (z)=q$
$\therefore \arg (\omega)=\left(\theta-\frac{3 \pi}{2}\right)$
$\text { So, } z=r e^{1 \theta}$
$\Rightarrow \bar{z}=r e^{i(-\theta)}$
$\omega=\frac{1}{r} e^{i\left(\theta-\frac{3 \pi}{2}\right)}$
Now, consider
$\frac{1-w \bar{z} \omega}{1+3 \bar{z} \omega}=\frac{1-2 e^{\left(-\frac{3 \pi}{2}\right)}}{1-3 e^{\left(-\frac{3 \pi}{2}\right)}}=\left(\frac{1-2 i}{1+3 i}\right)$
$\therefore \text { prin } \arg \left(\frac{1-2 \bar{z} \omega}{1+3 \bar{z} \omega}\right)$
$=\operatorname{prin} \arg \left(\frac{1-2 \bar{z} \omega}{1+3 \bar{z} \omega}\right)$
$=\left(-\frac{1}{2}(1+i)\right)$
$=-\left(\pi-\frac{\pi}{4}\right)=\frac{-3 \pi}{4}$
Similar Questions
Let $z$ be complex number satisfying $|z|^3+2 z^2+4 z-8=0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-$I$ to the correct entries in List-$II$.
| List-$I$ | List-$II$ |
| ($P$) $|z|^2$ is equal to | ($1$) $12$ |
| ($Q$) $|z-\bar{z}|^2$ is equal to | ($2$) $4$ |
| ($R$) $|z|^2+|z+\bar{z}|^2$ is equal to | ($3$) $8$ |
| ($S$) $|z+1|^2$ is equal to | ($4$) $10$ |
| ($5$) $7$ |
The correct option is:
