Question

If $z_1$, $z_2$ and $z_3$, $z_4$ are two pairs of conjugate complexnumbers, then $arg\left(\frac{z_1}{z_4}\right)+arg\left(\frac{z_2}{z_3}\right)$ equals

• A. 0
• B. $\frac{\pi }{2}$
• C. $\frac{3\pi }{2}$
• D. $\pi$

Answer 1

Answer: (A)

0

$Z_2={\overline{Z}}_1$
$Z_4={\overline{Z}}_3$
$\Rightarrow arg\left(\frac{Z_1}{Z_4}\right)+arg\left(\frac{Z_2}{Z_3}\right)$
$\Rightarrow arg\left(z_1\right)-arg\left(z_4\right)+arg\left(z_2\right)-arg\left(z_3\right)$
if $\Rightarrow arg\left(z_1\right)=\theta$ then $arg\left(z_2\right)=-\theta$
$\Rightarrow arg\left(z_3\right)=\varphi$ then $arg\left(z_4\right)=-\varphi$
$\Rightarrow \theta -\varphi -(-\theta )-(-\varphi )=0$