Question

# If $$z_1 , z_2$$ and $$z_3$$ are complex numbers such that $$| z_1 | = | z_2 | = | z_3 | = \left| \dfrac {1}{z_1} + \dfrac {1}{z_2} + \dfrac {1}{z_3} \right| = 1$$ then $$| z_1 + z_2 + z_3 |$$ is :

A
Equal to 1
B
Less than 1
C
Greater than 3
D
Equal to 3

Solution

## The correct option is A Equal to 1$$1 = \left| \dfrac {1}{z_1} + \dfrac {1}{z_2} + \dfrac {1}{z_3} \right| = \left| \dfrac {z_1\bar{z_1}}{z_1} + \dfrac {z_2\bar{z_2}}{z_2} + \dfrac {z_3\bar{z_3}}{z_3} \right|$$$$[ \because | \bar{z_1}^2 = 1 = z_1 \bar{z_1} ]$$$$= | \bar{z_1} + \bar{z_2} + \bar{z_3} | = | \overline{z_1 + z_2 + z_3 } | = | z_1 + z_2 + z_3 |$$ $$\quad \quad \quad [ \because | \bar{z_1} | = | z_1 | ]$$Mathematics

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