Area under the curveHard
Question
Given f(x) = x4 + αx2 + βx + γ (where α, β, γ are real numbers) and roots of f(x) = 0 are the eccentricities of the curves y = x tan θ - 
(where g ≠ 0, u ± 0 & θ ∈ (0, π/2)) and x2 - y2 = 9, then the value of (α + β + γ) is -
(where g ≠ 0, u ± 0 & θ ∈ (0, π/2)) and x2 - y2 = 9, then the value of (α + β + γ) is -Options
A.3
B.2
C.1
D.-1
Solution
Given curves are parabola and rectangular hyperbola
∴ Eccentricities are 1, √2 respectively
∵ Irrational roots occur in conjugate pair, thus
root of f(x) = 0 are 1, √2, - √2
but sum of roots of given equation = 0
⇒ Fourth root = - 1
∴ f(x) = (x + 1)(x - 1). (x - √2)(x + √2)
= (x2 - 1) (x2 - 2) = x4 - 3x2 + 2
∴ α = -3, β = 0, γ = 2
∴ (α + β + γ) = - 1
∴ Eccentricities are 1, √2 respectively
∵ Irrational roots occur in conjugate pair, thus
root of f(x) = 0 are 1, √2, - √2
but sum of roots of given equation = 0
⇒ Fourth root = - 1
∴ f(x) = (x + 1)(x - 1). (x - √2)(x + √2)
= (x2 - 1) (x2 - 2) = x4 - 3x2 + 2
∴ α = -3, β = 0, γ = 2
∴ (α + β + γ) = - 1
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