Question

Let $z_1,z_2$ be the roots of the equations $z^2+az+12=0and{z}_1,z_2$ form an equilateral triangle with origin. Then, the value of $\left|a\right|$ is

Answer 1

Calculating the value of $\left|a\right|$:

Given, the roots of the equation $z^2+az+12=0$ are $z_1$ and $z_2$.

Since, $z_1,z_2$ and the origin$(0,0)$ form an equilateral triangle:

$$\begin{gathered} {{\mathbf{z}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{{\mathbf{z}}_{\mathbf{2}}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}{{\mathbf{z}}^{\mathbf{2}}}_{\mathbf{3}}\mathbf{=}\mathbf{}{\mathbf{z}}_{\mathbf{1}}{\mathbf{z}}_{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}{\mathbf{z}}_{\mathbf{2}}{\mathbf{z}}_{\mathbf{3}}\mathbf{}\mathbf{+}\mathbf{}{\mathbf{z}}_{\mathbf{1}}{\mathbf{z}}_{\mathbf{3}} \\ \therefore {z_1}^2+{z_2}^2=z_1{z}_2\left[z_3=0\right] \\ \Rightarrow {\left(z_1+z_2\right)}^2=3z_1{z}_2 \\ \Rightarrow {(-a)}^2=3\left(12\right)\left[\because z_1+z_2=-a,z_1{z}_2=b\right] \\ \Rightarrow a^2=36 \\ \Rightarrow \left|a\right|=6 \end{gathered}$$

Therefore, the value of $\left|a\right|$ is $6$.