Question

Let $z,\omega$ be complex numbers such that $\bar{z}+i\omega=0$ and $arg\,(z\omega)=\pi$, Then arg $z$ equals :

• A. $\frac{\pi}{4}$
• B. $\frac{\pi}{2}$
• C. $\frac{3\pi}{4}$
• D. $\frac{5\pi}{4}$

Answer 1

Answer: (C)

$\bar{z}+i\bar{\omega}=0$
$\Rightarrow \overline{\bar{z}+i\bar{\omega}}=0$
$\Rightarrow z-i\omega=0$
$\Rightarrow arg \left(z\omega\right)=\pi$
$\Rightarrow arg\left(\frac{z^{2}}{i}\right)=\pi$
$\Rightarrow arg \left(z^{2}\right)-arg\left(i\right)=\pi$
$\Rightarrow 2\,arg\left(z\right)-\frac{\pi}{2}=\pi$
$\Rightarrow arg\left(z\right)-\frac{3\pi}{4}$