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Question
If z1, z2, z3 are complex numbers such that \[\left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1\] then find the value of \[\left| z_1 + z_2 + z_3 \right|\] .
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Solution
\[\left| z_1 + z_2 + z_3 \right| = \left| \frac{z_1 \bar{{z_1}}}{\bar{{z_1}}} + \frac{z_2 \bar{{z_2}}}{\bar{{z_2}}} + \frac{z_3 \bar{{z_3}}}{\bar{{z_3}}} \right|\]
\[ = \left| \frac{\left| z_1 \right|^2}{\bar{{z_1}}} + \frac{\left| z_2 \right|^2}{\bar{{z_2}}} + \frac{\left| z_3 \right|^2}{\bar{{z_3}}} \right|\]
\[ = \left| \frac{1}{\bar{{z_1}}} + \frac{1}{\bar{{z_2}}} + \frac{1}{\bar{{z_3}}} \right| [ \because \left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = 1]\]
\[ = \bar{\left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right|}\]
\[ = 1 \left[ \because \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1 \right]\]
Thus, the value of \[\left| z_1 + z_2 + z_3 \right|\] is 1.
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