Question

For any two complex number $z_1,z_2$ and any two real numbers $a$ and $b$,
$|az_1-bz_2|^2+|bz_1+az_2|^2=$

Answer 1

We know that,
$[|z_1-z_2|^2=|z_1|^2+|z_2|^2-2Re(z_1\overline{z_2})]$
and
$[|z_1+z_2|^2=|z_1|^2+|z_2|^2+2Re(z_1\overline{z_2})]$
$\therefore |az_1-bz_2|^2+|bz_1+az_2|^2=a^2|z_1|^2+b^2|z_2|-2Re(abz_1\overline{z_2})+b^2|z_1|^2+a^2|z_2|+2Re(abz_1\overline{z_2})$
$=|z_1|^2(a^2+b^2)+|z_2|^2(a^2+b^2)$
$=(a^2+b^2)(|z_1|^2+|z_2|^2)$