Question

Let $Z _{1}$ and $Z _{2}$ be two complex numbers such that $\overline{ z }_{1}= i \overline{ z }_{2}$ and $\arg \left(\frac{ z _{1}}{\overline{ z }_{2}}\right)=\pi$. Then

• A. $\arg z _{2}=\frac{\pi}{4}$
• B. $\arg z _{2}=-\frac{3 \pi}{4}$
• C. $\arg z _{1}=\frac{\pi}{4}$
• D. $\arg z _{1}=-\frac{3 \pi}{4}$

Answer 1

Answer: (C)

$\overline{ Z }_{1}= i \overline{ Z }_{2}$
$z _{1}=- iz _{2}$
$\arg \left(\frac{ Z _{1}}{ Z _{2}}\right)=\pi$
$\arg \left(- i \frac{ Z _{2}}{\overline{ Z }_{2}}\right)=\pi$
$\arg \left( z _{2}\right)=\theta$
$-\frac{\pi}{2}+\theta+\theta=\pi$
$2 \theta=\frac{3 \pi}{2}$
$\arg \left(z_{2}\right)=\theta=\frac{3 \pi}{4}, \arg z_{1}=\frac{\pi}{4}$