If $| z_{1} | = | z_{2} | = | z_{3} | = | z_{4} | = 1$ and $z_{1} + z_{2} + z_{3} + z_{4} = 0$ , then least value of the expression
$E = | z_{1} - z_{2} |^{2} + | z_{2} - z_{3} |^{2} + | z_{3} - z_{4} |^{2} + | z_{4} - z_{1} |^{2}$ is

6

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Solution

The correct option is B $8$
Given that, $| z_{1} | = | z_{2} | = | z_{3} | = | z_{4} | = 1$
and $z_{1} + z_{2} + z_{3} + z_{4} = 0$ ....(1)
$\Rightarrow_{1} +_{2} +_{3} +_{4} = 0$ ....(2)

Now, $| z_{1} - z_{2} |^{2} = (z_{1} - z_{2}) (_{1} -_{2}) = | z_{1} |^{2} + | z_{2} |^{2} - (z_{1}_{2} +_{1} z_{2})$

$E = | z_{1} - z_{2} |^{2} + | z_{2} - z_{3} |^{2} + | z_{3} - z_{4} |^{2} + | z_{4} - z_{1} |^{2}$
$\Rightarrow E = 2 (| z_{1} |^{2} + | z_{2} |^{2} + | z_{3} |^{2} + | z_{4} |^{2}) - (z_{1}_{2} +_{1} z_{2}) - (z_{1}_{2} +_{1} z_{2}) - (z_{1}_{2} +_{1} z_{2}) - (z_{1}_{2} +_{1} z_{2})$
$\Rightarrow E = 8 - [(z_{1} + z_{3}) (_{2} +_{4}) + (_{1} +_{3}) (z_{2} + z_{4})]$
$\Rightarrow E = 8 + [(z_{1} + z_{3}) (_{1} +_{3}) + (_{1} +_{3}) (z_{1} + z_{3})]$ ....[ from (1) & (2)]
$\Rightarrow E = 8 + 2 | z_{1} + z_{3} |^{2}$
Since, $| z_{1} + z_{3} |^{2} \geq 0$
Therefore, $E \geq 8$
Ans: B

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