Question

If the origin and the non-real roots of the equation $3z^{2}+3z+\lambda =0$ , $\forall \lambda \in R$ are the vertices of an equilateral triangle in the argand plane, then $\sqrt{3}$ times the length of the triangle is

• A. $2$ units
• B. $1$ unit
• C. $3$ units
• D. $4$ units

Answer 1

Answer: (B)

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