Question

α, β are roots of the equation λ (x² – x) + x + 5 = 0. If λ ₁ and λ ₂ are the two values of λ for which the roots α, β are connected by the relation $\frac{\alpha}{\beta}$ + $\frac{\beta}{\alpha}$ = 4, then the value of $\frac{\lambda_{1}}{\lambda_{2}}$ + $\frac{\lambda_{2}}{\lambda_{1}}$ is –

• A. 150
• B. 254
• C. 180
• D. 102

Answer 1

Answer: (D)

102

Answer 2

λ (x² – x) + x + 5 = 0 ⇒ λ x² + (1 – λ) x + 5 = 0

∴α + β = – $\frac{1 - \lambda}{\lambda}$ = $\frac{\lambda - 1}{\lambda}$, αβ = $\frac{5}{\lambda}$

Now $\frac{\alpha}{\beta}$+ $\frac{\beta}{\alpha}$= 4 ⇒ α² + β² = 4αβ

⇒ (α + β)² – 2 αβ = 4αβ

⇒ (α + β)² = 6 αβ ⇒ $\left( \frac{\lambda - 1}{\lambda} \right)^{2}$= 6$\left( \frac{5}{\lambda} \right)$

⇒ λ ² + 1 – 2 λ – 30 λ = 0

⇒ λ ² – 32λ + 1 = 0 

∴λ ₁ + λ ₂ = 32, λ ₁ λ ₂ = 1

Now $\frac{\lambda_{1}}{\lambda_{2}}$+ $\frac{\lambda_{2}}{\lambda_{1}}$= $\frac{\lambda_{1}^{2} + \lambda_{2}^{2}}{\lambda_{1}\lambda_{2}}$= $\frac{(32)^{2} - 2(1)}{1}$= 1022.