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Q.
If $z_1$ and $z_2$ are two complex numbers and $a, b$ are two real numbers, then $\left|a z_1-b z_2\right|^2+\left|b z_1+a z_2\right|^2$ equals
Complex Numbers and Quadratic Equations
Solution:
We have
$ \left|a z_1-b z_2\right|^2+\left|b z_1+a z_2\right|^2 $
$= a^2\left|z_1\right|^2+b^2\left|z_2\right|^2-a b z_1 \bar{z}_2-a b \bar{z}_1 z_2 +b^2\left|z_1\right|^2+a^2\left|z_2\right|^2+b a z \bar{z}_2+b a \bar{z}_1 z_2$
$= \left(a^2+b^2\right)\left(\left|z_1\right|^2+\left|z_2\right|^2\right)$