Question

Let $z_1,z_2$ be the roots of the equations $z^2+az+12=0$ and $z_1,z_2$ form an equilateral triangle with origin. Then, the value of $|a|$ is

Answer 1

In any equilateral triangle,
$z_1^2+z_2^2+z_3^2=z_1{z}_2+z_2{z}_3+z_3{z}_1$


$\Rightarrow z_1^2+z_2^2=z_1{z}_2(\because z_3=0)$
$\Rightarrow (z_1+z_2{)}^2=3z_1{z}_2$
$\Rightarrow a^2=36$
$\Rightarrow |a|=6$