A) equal to 1
B) less than 1
C) greater than 3
D) equal to 3
Correct Answer: A
Solution :
\[\left| {{z}_{1}} \right|\,=\,\,\left| {{z}_{2}} \right|=\left| {{z}_{3}} \right|=1\,\,\] (given) Now, \[\left| {{z}_{1}} \right|=1\,\,\Rightarrow \,\,{{\left| {{z}_{1}} \right|}^{2}}=1\,\,\Rightarrow \,\,{{z}_{1}}{{\bar{z}}_{1}}=1\] Similarly, \[{{\operatorname{z}}_{2}}{{\bar{z}}_{2}}=\,\,1,\,\,{{z}_{3}}{{\bar{z}}_{3}}=1\] Now, \[\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+\frac{1}{{{z}_{3}}} \right|=1\,\,\,\,\,\Rightarrow \,\,\,\left| {{{\bar{z}}}_{1}}+{{{\bar{z}}}_{2}}+{{{\bar{z}}}_{3}} \right|=1\] \[\Rightarrow \,\,\,\,|\overline{{{z}_{1}}+{{z}_{2}}+{{z}_{3}}}|=1\,\,\Rightarrow \,\,|{{z}_{1}}+{{z}_{2}}+{{z}_{3}}|=1\]
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