Let $z_{1}, z_{2}$ be the roots of the equation $z^{2}+a z+12=0$ and $z_{1}, z_{2}$ form an equilate

Question

Let $z_{1}, z_{2}$ be the roots of the equation $z^{2}+a z+12=0$ and $z_{1}, z_{2}$ form an equilateral triangle with origin. Then, the value of $|a|$ is

Professor PC Thomas Classes · Answer

(6): We have, $z_{1}+z_{2}=-a$ and\ $z_{1} z_{2}=12$\ $\because \quad z_{1}, z_{2}$ form an equilateral triangle with origin.\ $	herefore \quad z_{1}{ }^{2}+z_{2}{ }^{2}=z_{1} z_{2}$\ Now, $\left(z_{1}+z_{2}ight)^{2}=z_{1}^{2}+z_{2}^{2}+2 z_{1} z_{2}=3 z_{1} z_{2}$\ $\Rightarrow(-a)^{2}=3 	imes 12=36 \Rightarrow a= \pm 6$\ $\Rightarrow \quad|a|=6$\