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# Let $z,\omega$ be two comlex numbers such that $\overline{z}+i\overline{\omega}=0$ and $arg(z\omega)=\pi$, then $arg(z)$ equal to

$\begin{array}{1 1}(A)\large\frac{\pi}{4} \\ (B) \large\frac{\pi}{2} \\ (C) \large\frac{3 \pi}{4}\\(D) \large\frac{5 \pi}{4} \end{array}$

Can you answer this question?

$\overline{z}+i\overline{\omega}=0$
Taking conjugate $\Rightarrow z-i\omega=0$
$arg(z\omega)=\pi$
$\Rightarrow arg(\large\frac{z^2}{i})=\pi$
$\Rightarrow 2arg(z)-arg(i)=\pi$
$\Rightarrow 2arg(z)=\pi+\large\frac{\pi}{2}=\frac{3\pi}{2}$
$\Rightarrow arg(z)=\large\frac{3\pi}{4}$
Hence (C) is the correct answer.
answered Apr 9, 2014