Question

Let $z,w$ be complex numbers such that $\overline{z}+i\overline{w}=0$ and $\arg zw=\pi$. Then $\arg z$ equals

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

Answer 1

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

$\overline{z}+i\overline{w}=0$
$\Rightarrow z-iw=0\Rightarrow z=iw\dots \left[1\right]$
$\arg zw=\pi$
$\Rightarrow \arg z+\arg w=\pi$
$\Rightarrow \arg z+\arg \frac{z}{i}=\pi$
$\Rightarrow 2\arg z-\arg i=\pi$
$\Rightarrow 2\arg z-\frac{\pi }{2}=\pi$
$\Rightarrow \arg z=\frac{3\pi }{4}$