Question

If $z_1,z_2$ and $z_3,z_4$ are two pairs of conjugate complex numbers, 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

The correct option is A $0$

Given $z_1$ & $z_2$ & $z_3,z_4$ are conjugate complete no ;

$\therefore z_2=\overline{z_1}$ & $z_4=\overline{z_3}$

$arg\left(\frac{z_1}{z_4}\right)+arg\left(\frac{z_2}{z_3}\right)=arg\left(\frac{z_1}{z_4}\right)\left(\frac{z_2}{z_3}\right)$

$=arg\left(\frac{z_1}{z_3}\right)\left(\frac{\overline{z_1}}{z_3}\right)$

$=arg\left(\frac{z_1\overline{z_1}}{z_3\overline{z_3}}\right)$

$=arg\left(\frac{{\left|z_1\right|}^2}{{\left|z_3\right|}^2}\right)$

$=0(\frac{{\left|z_1\right|}^2}{{\left|z_3\right|}^2}\Rightarrow$ purely real )