Question

If $\lambda \in R$ such that the origin and the non-real roots of the equation $2z^{2}+2z+\lambda =0$ form the vertices of an equilateral triangle in the argand plane, then $\frac{1}{\lambda }$ is equal to

Answer 1

Let, $z_{1}$ and $z_{2}$ are roots of the equation $2z^{2}+2z+\lambda =0$
$\Rightarrow z_{1}+z_{2}=-1andz_{1}z_{2}=\frac{\lambda }{2}$
If $0,z_{1}\&z_{2}$ form an equilateral triangle, then $z_{1}^{2}+z_{2}^{2}=z_{1}z_{2}$
$\Rightarrow \left(z_{1} + z_{2}\right)^{2}=3z_{1}z_{2}$
$\Rightarrow \left(- 1\right)^{2}=3\left(\frac{\lambda }{2}\right)\Rightarrow \frac{1}{\lambda }=\frac{3}{2}=1.5$