If the origin and the non-real roots of the equation 3z2+3z+λ =0 , ∀ λ ∈ R are the vertices of an equilateral triangle in the argand plane, then √3 times the length of the triangle is

Question

If the origin and the non-real roots of the equation 3z2+3z+λ =0 , ∀ λ ∈ R are the vertices of an equilateral triangle in the argand plane, then √3 times the length of the triangle is (A) 2 units (B) 1 unit (C) 3 units (D) 4 units

Tardigrade Admin · Accepted Answer

Let, z1 and z2 are roots of the equation 3z2+3z+λ =0 ⇒ z1+z2=-1 and z1z2=(λ /3) If 0,z1 and z2 form an equilateral triangle, then z12+z22=z1z2 ⇒ (z1 + z2)2=3z1z2 ⇒ (- 1)2=3((λ /3))⇒ λ =1⇒ |z1 z2|=(1/3) ⇒ √3|z1|=1 unit

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

$2$ units

$1$ unit

$3$ units

$4$ units

Q. If the origin and the non-real roots of the equation 3z2+3z+λ=0 , ∀λ∈R are the vertices of an equilateral triangle in the argand plane, then 3​ times the length of the triangle is

2 units

1 unit

3 units

4 units

Let, z1​ and z2​ are roots of the equation 3z2+3z+λ=0 ⇒z1​+z2​=−1 and z1​z2​=3λ​ If 0,z1​ and z2​ form an equilateral triangle, then z12​+z22​=z1​z2​ ⇒(z1​+z2​)2=3z1​z2​ ⇒(−1)2=3(3λ​)⇒λ=1⇒∣z1​z2​∣=31​ ⇒3​∣z1​∣=1 unit

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

$2$ units

$1$ unit

$3$ units

$4$ units