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