Explanation

(a)
$\mathcal{z}_{2} + ax + a^{2} = 0\ \Rightarrow \ \mathcal{z =}a\omega,\ a\omega^{2}$
(where $'\omega'$ is a non-real root of unity)
$\Rightarrow \$ Locus of $\mathcal{z}$ is a pair of straight lines
and $\arg\ \left( \mathcal{z} \right) = \arg{(a) + \arg{(\omega)}}$
or $\arg{\left( \mathcal{z} \right) = \arg{(a) + \arg{(\omega^{2})}}}$
$\Rightarrow \ \arg{\left( \mathcal{z} \right) = \pm \frac{2\pi}{3}}$
Also, $\left| \mathcal{z} \right| = |a||\omega|\ \ or\ \ \left| \mathcal{z} \right| = |a||\omega^{2}|$
$\Rightarrow \ \ \left| \mathcal{z} \right| = |a|$