• # question_answer If ${{z}_{1}},{{z}_{2}},{{z}_{3}}$ are complex numbers such that $|{{z}_{1}}|\,=\,|{{z}_{2}}|\,=$ $\,|{{z}_{3}}|\,=$ $\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+\frac{1}{{{z}_{3}}} \right|=1\,,$ then$\text{ }|{{z}_{1}}+{{z}_{2}}+{{z}_{3}}|$ is [MP PET 2004; IIT Screening 2000] A) Equal to 1 B) Less than 1 C) Greater than 3 D) Equal to 3

$1=\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+\frac{1}{{{z}_{3}}} \right|$$=\left| \frac{{{z}_{1}}{{{\bar{z}}}_{1}}}{{{z}_{1}}}+\frac{{{z}_{2}}{{{\bar{z}}}_{2}}}{{{z}_{2}}}+\frac{{{z}_{3}}{{{\bar{z}}}_{3}}}{{{z}_{3}}} \right|$$(\because \,\,\,|{{z}_{1}}{{|}^{2}}=1={{z}_{1}}{{\overline{z}}_{1}},\text{etc})$ $=\,|{{\bar{z}}_{1}}+{{\bar{z}}_{2}}+{{\bar{z}}_{3}}|\,=\,|\overline{{{z}_{1}}+{{z}_{2}}+{{z}_{3}}}|\,=\,|{{z}_{1}}+{{z}_{2}}+{{z}_{3}}|$ $(\because \,\,\,|{{\bar{z}}_{1}}|=|{{z}_{1}}|)$