$\left|z_1+z_2\right|^2+\left|z_1-z_2\right|^2$ is equal to
$\left|z_1+z_2\right|^2+\left|z_1-z_2\right|^2$ is equal to
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$2\left(\left|z_1\right|+\left|z_2\right|\right.$
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$2\left(\left|z_1\right|^2+\left|z_2\right|^2\right)$
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$\left|z_1\right|\left|z_2\right|$
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$\left|z_1\right|^2+\left|z_2\right|^2$
Solution
$\left|z_1+z_2\right|^2+\left|z_1-z_2\right|^2$
$
=\left|z_1\right|^2+\left|z_2\right|^2+2\left|z_1\right|\left|z_2\right|+\left|z_1\right|^2+\left|z_2\right|^2-2\left|z_1\right|\left|z_2\right|
$
$
=2\left|z_1\right|^2+2\left|z_2\right|^2=2\left[\left|z_1\right|^2+\left|z_2\right|^2\right]
$
Asked in: JEE Main 2012 (26 May Online)
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