A) Equal to 1
B) Less than 1
C) Greater than 3
D) Equal to 3
Correct Answer: A
Solution :
\[1=\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+\frac{1}{{{z}_{3}}} \right|\]\[=\left| \frac{{{z}_{1}}{{{\bar{z}}}_{1}}}{{{z}_{1}}}+\frac{{{z}_{2}}{{{\bar{z}}}_{2}}}{{{z}_{2}}}+\frac{{{z}_{3}}{{{\bar{z}}}_{3}}}{{{z}_{3}}} \right|\]\[(\because \,\,\,|{{z}_{1}}{{|}^{2}}=1={{z}_{1}}{{\overline{z}}_{1}},\text{etc})\] \[=\,|{{\bar{z}}_{1}}+{{\bar{z}}_{2}}+{{\bar{z}}_{3}}|\,=\,|\overline{{{z}_{1}}+{{z}_{2}}+{{z}_{3}}}|\,=\,|{{z}_{1}}+{{z}_{2}}+{{z}_{3}}|\] \[(\because \,\,\,|{{\bar{z}}_{1}}|=|{{z}_{1}}|)\]
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