Let (α, β), (β, γ) and (γ, α) are respectively the roots of x2 – 2px + 2 = 0, x2 – 2qx + 3 = 0 and x2 – 2rx + 6 = 0. If α, β and γ are all positive then the value of (p + q + r) is

Engineering

Mathematics

Quadratic Equation and Nature of Roots

Question

Let (α, β), (β, γ) and (γ, α) are respectively the roots of x² – 2px + 2 = 0, x² – 2qx + 3 = 0 and x² – 2rx + 6 = 0. If α, β and γ are all positive then the value of (p + q + r) is

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$∵$ α + β = 2p, b + γ = 2q and γ + α = 2r
⇒ α + β + γ = p + q + r
and αβ = 2, βγ = 3, γα = 6
$∴$ αβγ = 6
⇒ γ = 3, α = 2, β = 1
⇒ p + q + r = 6.