For any two complex numbers $z_{1}, z_{2}$ and any real numbers a & b; $| a z_{1} - b z_{2} |^{2} + | b z_{1} + a z_{2} |^{2} = k (a^{2} + b^{2}) (| z_{1} |^{2} + | z_{2} |^{2})$ . What is the value of $2 k$ ?

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Solution

$| a z_{1} - b z_{2} |^{2} + | b z_{1} + a z_{2} |^{2} = (a z_{1} - b z_{2}) (a_{1} - b_{2}) + (a z_{1} + b z_{2}) (a_{1} + b_{2})$
$= a^{2} {| z_{1} |}^{2} - a b z_{1}_{2} - a b z_{2}_{1} + b^{2} {| z_{2} |}^{2} + a^{2} {| z_{1} |}^{2} + a b z_{1}_{2} + a b z_{2}_{2} + b^{2} {| z_{2} |}^{2}$
$\Rightarrow | a z_{1} - b z_{2} |^{2} + | b z_{1} + a z_{2} |^{2} = (a^{2} + b^{2}) (| z_{1} |^{2} + | z_{2} |^{2})$

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