Question

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

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

Answer 1

Answer: (C)

$\frac{3\pi }{4}$

Let $z=r{e}^{i\theta }$ and $w=r^{\prime }{e}^{i\alpha }$
Hence, $\overline{z}+\overline{iw}=0$
$r{e}^{-i\theta }=-i.r^{\prime }\left(e^{-i\alpha }\right)$
$r{e}^{-i\theta }=r^{\prime }\left(e^{-(i\alpha +\frac{\pi }{2})}\right)$
Hence, $r=r^{\prime }$ and
$\theta =\alpha +\frac{\pi }{2}$.
Hence $z=r{e}^{i\theta }$ and $w=r{e}^{i(\theta -\frac{\pi }{2})}$
Now $z.w$ $=r^2.e^{i(2\theta -\frac{\pi }{2})}$
Now $2\theta -\frac{\pi }{2}=\pi$
$2\theta =\frac{3\pi }{2}$
$\therefore \theta =\frac{3\pi }{4}$
Hence, $arg\left(z\right)=\frac{3\pi }{4}$.