If w h e r e are the roots of the equation x 2 - ( + 3 ) x + 3 = 0 such that 1 - 1 = 1 3 , then the sum of all possible values of is [2026]

Question

If $α, β where α < β$ are the roots of the equation $λ x^{2} - (λ + 3) x + 3 = 0$ such that $\frac{1}{α} - \frac{1}{β} = \frac{1}{3},$ then the sum of all possible values of $λ$ is [2026]

Smart Achievers · Accepted Answer

(3)

$\frac{β - α}{α β} = \frac{1}{3}, α + β = \frac{λ + 3}{λ}, α β = \frac{3}{λ}$

$β - α = \frac{α β}{3} = \frac{1}{λ}$

$on squaring$

$α^{2} + β^{2} - 2 α β = \frac{1}{λ^{2}} . . . (1)$

$α^{2} + β^{2} + 2 α β = \frac{{(λ + 3)}^{2}}{λ^{2}} . . . (2)$

(2) - (1) $4 α β = \frac{{(λ + 3)}^{2} - 1}{λ^{2}}$

$\frac{12}{λ} = \frac{λ^{2} + 6 λ + 8}{λ^{2}}$

$\Rightarrow λ^{2} - 6 λ + 8 λ = 0$

$\Rightarrow λ = 0, 2, 4$

$Sum of possible values of λ is 6$

Solution

Q.
If $α, β where α < β$ are the roots of the equation $λ x^{2} - (λ + 3) x + 3 = 0$ such that $\frac{1}{α} - \frac{1}{β} = \frac{1}{3},$ then the sum of all possible values of $λ$ is [2026]

1 8

2 2

3 6

4 4

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Solution

Q. If α,β where α<β are the roots of the equation λx2-(λ+3)x+3=0 such that 1α-1β=13, then the sum of all possible values of λ is [2026]

1 8

2 2

3 6

4 4

Ans. (3) β-ααβ=13, α+β=λ+3λ, αβ=3λ β-α=αβ3=1λ on squaring α2+β2-2αβ=1λ2 ...(1) α2+β2+2αβ=(λ+3)2λ2 ...(2) (2) - (1) 4αβ=(λ+3)2-1λ2 12λ=λ2+6λ+8λ2 ⇒λ2-6λ+8λ=0 ⇒λ=0,2,4 Sum of possible values of λ is 6

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Q.
If $α, β where α < β$ are the roots of the equation $λ x^{2} - (λ + 3) x + 3 = 0$ such that $\frac{1}{α} - \frac{1}{β} = \frac{1}{3},$ then the sum of all possible values of $λ$ is [2026]