A) 0
B) \[\frac{\pi }{2}\]
C) \[\frac{3\pi }{2}\]
D) \[\pi \]
Correct Answer: A
Solution :
Consider \[\arg \left( \frac{{{z}_{1}}}{{{z}_{4}}} \right)+\arg \left( \frac{{{z}_{2}}}{{{z}_{3}}} \right)\] \[=\arg ({{z}_{1}})-arg({{z}_{4}})+arg({{z}_{2}})-arg({{z}_{3}})\] \[=\arg ({{z}_{1}})+arg({{z}_{2}}))-(arg({{z}_{3}})+arg({{z}_{4}}))\] given\[\left( \begin{matrix} {{z}_{2}}={{\overline{z}}_{1}}\And \\ {{z}_{4}}={{\overline{z}}_{3}} \\ \end{matrix} \right)\] \[=(\arg \,({{z}_{1}})+arg({{\overline{z}}_{1}}))-(arg({{z}_{3}})+arg({{\overline{z}}_{3}}))\] \[\left\{ \begin{align} & also\,(\arg ({{\overline{z}}_{1}})=-arg({{z}_{1}}) \\ & arg({{\overline{z}}_{3}})=-arg({{z}_{3}}) \\ \end{align} \right\}\] \[=\,(\arg \,({{z}_{1}})-arg({{z}_{1}}))-(arg({{z}_{3}})-arg({{z}_{3}}))\] \[=0-0=0\]
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