Question

Let $z$ and $\omega$ be complex numbers such that $\overline{z}+i\overline{\omega }=0$ and $arg\left(z\omega \right)=\pi$. Then, $arg\left(z\right)$ is equal to

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

Answer 1

The correct option is C $\frac{3\pi }{4}$

Since, $\overline{z}+i\overline{\omega }=0$
$\Rightarrow \overline{z}=-i\overline{\omega }$
$\Rightarrow z=i\omega$
$\Rightarrow \omega =-iz$
Also, $arg\left(z\omega \right)=\pi$
$\Rightarrow arg(-i{z}^2)=\pi$
$\Rightarrow arg(-i)+2arg\left(z\right)=\pi$
......$[\because arg(-i)=-\frac{\pi }{2}]$
$\Rightarrow -\frac{\pi }{2}+2arg\left(z\right)=\pi$
$\Rightarrow 2arg\left(z\right)=\frac{3\pi }{2}$
$\Rightarrow arg\left(z\right)=\frac{3\pi }{4}$