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)

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