Question

The roots of the equation $z^4+a{z}^3+(12+9i)z^2+bz=0$ (where a and b are complex numbers ) are the vertices of a square, then the value of $|a{|}^2$ is

Answer 1

z=0

$z^3+a{z}^2+(12+9i)z+b=0$

Let the vertices of a square be $0,u,iu,u+ui$


$u.\left(ui\right)+iu(u+ui)+(u+ui).u=12+9i$

$\left(3i{u}^2\right)=12+9i\Rightarrow u^2=3-4i$

$u=\sqrt{3-4i}=\pm (2-i)$

$u=2-i$

so $-a=u+ui+u(1+i)$

$-a=2u(1+i)$

$a=-6-2i$
$|a|=\sqrt{36+4}=\sqrt{40}$
$\therefore |a{|}^2=40$