If - are the roots of the equation x2-4x-0 and - are the roots of the equation x2-64x-0 and - forms an increasing G-P- then the value of - is

Assuming nbsp;α= a,β = ar,γ = ar^2,δ = ar^3 x^2 4x+λ=0 So, α+β = 4 α·β = λ x^2 64x+μ=0 So, nbsp; γ+δ = 64 γ·δ = μ Now, nbsp; γ+δα+β = 16 ⇒ r^2 = 16⇒ r = ± ...

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Standard XIII

Mathematics

Sum of Infinite Terms

If α,β are th...

Question

If $α, β$ are the roots of the equation $x^{2} - 4 x + λ = 0$ and $γ, δ$ are the roots of the equation $x^{2} - 64 x + μ = 0$ and $α, β, γ, δ$ forms an increasing G.P., then the value of $\frac{μ}{λ}$ is

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Solution

Assuming $α = a, β = a r, γ = a r^{2}, δ = a r^{3}$
$x^{2} - 4 x + λ = 0$
So,
$α + β = 4 α \cdot β = λ$
$x^{2} - 64 x + μ = 0$
So,
$γ + δ = 64 γ \cdot δ = μ$
Now,
$\frac{γ + δ}{α + β} = 16 \Rightarrow r^{2} = 16 \Rightarrow r = \pm 4$
Therefore,
$\frac{μ}{λ} = \frac{γ \cdot δ}{α \cdot β} \Rightarrow \frac{μ}{λ} = r^{4} ∴ \frac{μ}{λ} = 256$

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If α,β are the roots of the equation x2−4x+λ=0 and γ,δ are the roots of the equation x2−64x+μ=0 and α,β,γ,δ forms an increasing G.P., then the value of μλ is

Assuming α=a,β=ar,γ=ar2,δ=ar3 x2−4x+λ=0 So, α+β=4α⋅β=λ x2−64x+μ=0 So, γ+δ=64γ⋅δ=μ Now, γ+δα+β=16⇒r2=16⇒r=±4 Therefore, μλ=γ⋅δα⋅β⇒μλ=r4∴μλ=256

If $α, β$ are the roots of the equation $x^{2} - 4 x + λ = 0$ and $γ, δ$ are the roots of the equation $x^{2} - 64 x + μ = 0$ and $α, β, γ, δ$ forms an increasing G.P., then the value of $\frac{μ}{λ}$ is