If - - are the roots of x2-3x-a-0 and - - that of x2-12x-b-0 and - - - - form an increasing G-P- then

The correct option is D a=2, b=32 αβ=a #160; #160; ; #160; #160; α + β=3 γδ =b #160; #160; ; #160; #160; γ +δ =12. SInce α ...

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Standard X

Mathematics

Properties of GP

If α, β are...

Question

If $α, β$ are the roots of $x^{2} - 3 x + a = 0$ and $γ, δ$ that of $x^{2} - 12 x + b = 0$ and $α, β, γ, δ$ form an increasing G.P, then

a = 3, b = 12

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a = 4, b = 16

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a = 2, b = 32

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a = 12, b = 3

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Solution

The correct option is D $a = 2, b = 32$
$α β = a; α + β = 3$
$γ δ = b; γ + δ = 12.$
SInce $α, β, γ, δ$ are in G.P
$\Rightarrow \frac{α}{β} = \frac{γ}{δ}$
$\frac{α + β}{α - β} = \frac{γ + δ}{γ - δ} .$
$\frac{3}{\sqrt{3^{2} - 4 a}} = \frac{12}{\sqrt{12^{2} - 4 b}}$
$∴ b = 16 a$
$\Rightarrow b = 32, a = 2$

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If α, β are the roots of x2−3x+a=0 and γ, δ that of x2−12x+b=0 and α, β, γ, δ form an increasing G.P, then

The correct option is D a=2, b=32αβ=a;α+β=3γδ=b;γ+δ=12.SInce α,β,γ,δ are in G.P⇒αβ=γδα+βα−β=γ+δγ−δ.3√32−4a=12√122−4b∴b=16a⇒b=32,a=2

If $α, β$ are the roots of $x^{2} - 3 x + a = 0$ and $γ, δ$ that of $x^{2} - 12 x + b = 0$ and $α, β, γ, δ$ form an increasing G.P, then