For any two complex numbers $z_{1}, z_{2}$ and any two real numbers a and b,
$| a z_{1} - b z_{2} |^{2} + | b z_{1} + a z_{2} |^{2} = . . . . .$ .

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Solution

$| a z_{1} - b z_{2} |^{2} + | b z_{1} + a z_{2} |^{2}$
$= | a z_{1} |^{2} | + | b z_{2} |^{2} - 2 | a b z_{1} z_{2} | + | b z_{1} |^{2} + | a z_{2} |^{2} + 2 | a z_{1} b z_{2} |$
$= a^{2} | z_{1} |^{2} + b^{2} | z_{2} |^{2} + b^{2} | z_{1} |^{2} + a^{2} (| z_{2} |)^{2}$
$= a^{2} (| z_{1} |^{2} + | z_{2} |^{2}) + b^{2} (| z_{1} |^{2} + | z_{2} |^{2})$
$= (a^{2} + b^{2}) (| z_{1} |^{2} + | z_{2} |^{2})$ .

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For any two complex numbers z1,z2 and any two real numbers a and b,|az1−bz2|2+|bz1+az2|2=..... .

|az1−bz2|2+|bz1+az2|2=|az1|2|+|bz2|2−2|abz1z2|+|bz1|2+|az2|2+2|az1bz2|=a2|z1|2+b2|z2|2+b2|z1|2+a2(|z2|)2=a2(|z1|2+|z2|2)+b2(|z1|2+|z2|2)=(a2+b2)(|z1|2+|z2|2).

For any two complex numbers $z_{1}, z_{2}$ and any two real numbers a and b,
$| a z_{1} - b z_{2} |^{2} + | b z_{1} + a z_{2} |^{2} = . . . . .$ .