If the origin and the non-real roots of the equation 3 z 2 + 3 z + λ = 0 , ∀ λ ∈ R are the vertices of an equilateral triangle in the argand plane, then 3 times the length of the triangle is

Step-by-step solution

Let, z 1 and z 2 are roots of the equation 3 z 2 + 3 z + λ = 0 ⇒ z 1 + z 2 = - 1 and z 1 z 2 = λ 3 If 0 ,   z 1 and z 2 form an equilateral triangle, then z 1 2 + z 2 2 = z 1 z 2 ⇒ z 1 + z 2 2 = 3 z 1 z 2 ⇒ - 1 2 = 3 λ 3 ⇒ λ = 1 ⇒ z 1 z 2 = 1 3 ⇒ 3 z 1 = 1 unit