Equations x3 + 4x2 + px + q = 0 and x3 + 9x2 + px + r = 0 are two roots in common, their third roots are γ1 & γ2 respectively. If x, y, z be three positive real numbers in G.P. such that | γ1 | is A.M. between x & y and 9 is H.M. between y and z, then y is equal to

Engineering

Mathematics

cubic and higher degree equation

Question

Equations x³ + 4x² + px + q = 0 and x³ + 9x² + px + r = 0 are two roots in common, their third roots are γ₁ & γ₂ respectively. If x, y, z be three positive real numbers in G.P. such that | γ₁ | is A.M. between x & y and 9 is H.M. between y and z, then y is equal to

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Let common roots be α and β

$∴$ αβ + βγ₁ + γ₁α = p = αβ + βγ₂ + γ₂α

⇒ (α + β) (γ₁ – γ₂) = 0

$∴$ α + β = 0

$∴$ α + β + γ₁ = – 4 ⇒ γ₁ = – 4 and γ₂ = – 9

Let x, y, z be $\frac{y}{r},$ y, yr where r is common ratio

$∴ \frac{\frac{y}{r} + y}{2} = 4 \Rightarrow \frac{y + yr}{r} = 8$ ….…(1)

$and \frac{2 y \cdot yr}{y + yr} = 9 \Rightarrow \frac{y^{2} r}{y + yr} = \frac{9}{2}$ ….…(2)

(1) × (2) ⇒ y² = 36 ⇒ y = 6.