Question

If $z_1,z_2$ and $z_3,z_4$ are two pairs of conjugate complex numbers, then the value of $\arg \left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right)$ is

• A. $\frac{\pi }{2}$
• B. $0$
• C. $\frac{\pi }{4}$
• D. $-\frac{\pi }{4}$

Answer 1

The correct option is B $0$

Given, $z_2={\overline{z}}_1$ and $z_4={\overline{z}}_3$
$\therefore z_1{z}_2=|z_1{|}^2$ and $z_3{z}_4=|z_3{|}^2$

Now, $\arg \left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right)$
$=\arg \left(\frac{z_1{z}_2}{z_4{z}_3}\right)$
$=\arg \left(\frac{|z_1{|}^2}{|z_3{|}^2}\right)$
$=\arg \left({\left|\frac{z_1}{z_3}\right|}^2\right)$
$=\arg \left(a\right),a\in R^{+}$
$=0$