If α, β are the roots of λx² − (λ + 3)x + 3 = 0 such that 1/α − 1/β = 1/3, find the sum of all possible values of λ

Q. If α, β, where α < β, are the roots of the equation λx² − (λ + 3)x + 3 = 0 such that 1/α − 1/β = 1/3, then the sum of all possible values of λ is

(A) 2

(B) 6

(D) 4

Correct Answer: 6

Explanation

For the quadratic equation

λx² − (λ + 3)x + 3 = 0

Sum and product of roots are:

α + β = (λ + 3)/λ, αβ = 3/λ

Given:

1/α − 1/β = (β − α)/(αβ) = 1/3

So,

β − α = (αβ)/3 = (3/λ)/3 = 1/λ

Now,

(β − α)² = (α + β)² − 4αβ

Substitute values:

(1/λ)² = ((λ + 3)/λ)² − 12/λ

Simplifying, we get:

λ² − 6λ + 8 = 0

Solving,

λ = 2, 4

Hence, the sum of all possible values of λ is:

2 + 4 = 6