• # question_answer 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) $\pi$

We have ${{z}_{2}}={{\overline{z}}_{1}}$and ${{z}_{4}}={{\overline{z}}_{3}}$ Therefore ${{z}_{1}}{{z}_{2}}=|{{z}_{1}}{{|}^{2}}$and  ${{z}_{3}}{{z}_{4}}=|{{z}_{3}}{{|}^{2}}$ Now $arg\left( \frac{{{z}_{1}}}{{{z}_{4}}} \right)+arg\left( \frac{{{z}_{2}}}{{{z}_{3}}} \right)=arg\left( \frac{{{z}_{1}}{{z}_{2}}}{{{z}_{4}}{{z}_{3}}} \right)$ $=arg\left( \frac{|{{z}_{1}}{{|}^{2}}}{|{{z}_{3}}{{|}^{2}}} \right)=arg\left( {{\left| \frac{{{z}_{1}}}{{{z}_{3}}} \right|}^{2}} \right)$= 0 ($\because$Argument of positive real number is zero).