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If $z_{1}, z_{2}$ and $z_{3}$ are complex numbers such that $|z_{1}| = |z_{2}| = |z_{3}| = 1 and z_{1} + z_{2} + z_{3} = 0$ then $|\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}|$ is

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equal to O

equal to 1

greater than 1 and less than 3

equal to 3

answer is A.

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Detailed Solution

$\begin{aligned} |z_{1}| = |z_{2}| = |z_{3}| = 1 \\ ∴ {\bar{z}}_{1} = \frac{1}{z_{1}}, {\bar{z}}_{2} = \frac{1}{z_{2}} \end{aligned}$

and ${\bar{z}}_{3} = \frac{1}{z_{3}} (∴ \bar{zz} = | z |^{2})$

$\begin{aligned} ∴ |\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}| = |{\bar{z}}_{1} + {\bar{z}}_{2} + {\bar{z}}_{3}| \\ = |z_{1} + z_{2} + z_{3}| = |z_{1} + z_{2} + z_{3}| = 0 \end{aligned}$

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