If $z_{1} + z_{2} + z_{3} + z_{4} = 0$ , then the expression $| z_{1} - z_{2} |^{2} + | z_{2} - z_{3} |^{2} + | z_{3} - z_{4} |^{2} + | z_{4} - z_{1} |^{2} - 2 (| z_{1} |^{2} + | z_{2} |^{2} + | z_{3} |^{2} + | z_{4} |^{2})$ is equal to $0$ , is and only if,

z_{1} = - z_{3}

and

z_{4} = - z_{2}

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z_{1} = - z_{4}

and

z_{2} = - z_{3}

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z_{1} = z_{2}

and

z_{3} = - z_{4}

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z_{1} = z_{3}

and

z_{2} = z_{4}

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Solution

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

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

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