Question

Let z, $\omega$ be complex numbes such that
$\overline{z}+i\overline{\omega }=0$ and $\arg \left(z\omega \right)=\pi$, then $\arg z=$

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

Answer 1

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

from the figure.
$z=|z|(cos\theta +sin\theta )--\left(1\right)$
$\omega =|\omega |(sin\theta -icos\theta )--\left(2\right)$
and $arg\left(z\omega \right)=\pi$
from complex numbers rule,
$cos{\theta }_1=\frac{z.\omega }{|z||\omega |}=\frac{|z|(cos\theta +isin\theta )|\omega |(sin\theta -icos\theta )}{|z||\omega |}$
as $arg\left(z\omega \right)=\pi ,\therefore {\theta }_1=\pi$
$cos{\theta }_1=-1=2cos\theta sin\theta$
$sin2\theta =-1$
$\Rightarrow 2\theta =\frac{3\pi }{2}$
$\Rightarrow \theta =\frac{3\pi }{4}$ option c