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

Answer 1

We have $z_{2}and{\overline{z}}_{1}and{z}_4={\overline{z}}_3$
Therefore $z_1{z}_2=|z_1{|}^{2}and{z}_3{z}_4=|z_3{|}^2$
Now $arg\left(\frac{z_1}{z_4}\right)+arg\left(\frac{z_2}{z_3}\right)=arg\left(\frac{z_1{z}_2}{z_4{z}_3}\right)$
$=arg\left(\frac{|z_1{|}^2}{|z_3{|}^2}\right)=arg\left(|\frac{z_1}{z_3}|\right)=0$

($\because$ Argument of positive real number is zero).