Question

For any two complex numbers $z_1$ and $z_2,$ if $|z_1|=2$ and $|z_2|=3$ then the value of $|3z_1+2z_2{|}^2+|3z_1-2z_2{|}^2$ is

Answer 1

$|3z_1+2z_2{|}^2+|3z_1-2z_2{|}^2$
$\left[\begin{array}{c}\because |z_1-z_2{|}^2=|z_1{|}^2+|z_2{|}^2-2Re\left(z_1\overline{z_2}\right),\\ \because |z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2+2Re\left(z_1\overline{z_2}\right)\end{array}\right]$
$=9|z_1{|}^2+4|z_2{|}^2+Re\left(6z_1{z}_2\right)+9|z_1{|}^2+4|z_2{|}^2-Re\left(6z_1{z}_2\right)$
$=18|z_1{|}^2+8|z_2{|}^2$
$=2(|3z_1{|}^2+|2z_2{|}^2)$
$\because |z_1|=2,|z_2|=3$
$\therefore 18|z_1{|}^2+8|z_2{|}^2=18\times 2^2+8\times 3^2=144$