Question

Let $z,w$ be two non-zero complex numbers. If $\overline{z+iw}=0$ and $arg\left(zw\right)=\pi$ then $argz$ is equal to

• A. $\pi$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{6}$
• E. $\frac{\pi }{8}$

Answer 1

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

$\overline{z+iw}=0\Rightarrow \overline{\overline{z+iw}}=\overline{0}$
$\Rightarrow z+iw=0\Rightarrow z=-iw$
$\because arg\left(zw\right)=\pi$
$arg\left(z\right)+arg\left(w\right)=\pi$
$[\because arg(z_1.z_2)=arg(z_1)+arg(z_2\left)\right]$
$\Rightarrow arg\left(z\right)+arg\left(z\right)+\frac{\pi }{2}=\pi$
$\Rightarrow arg\left(z\right)=\frac{\pi }{4}$