First slide

Geometric progression

Question

If $α$ and $β$ are roots of the equation $x^{2} - 3 x + a = 0$ and $γ$ and $δ$ are roots of the equation $x^{2} - 12 x + b = 0$ and $α, β, γ, δ$ form an increasing GP, then the values of a and b are respectively

Moderate

A

2,16
B

4,8
C

2,32
D

None of these

Solution

$∵ α, β, γ, δ$ are in GP.
Let $α = A, β = Ar, γ = {Ar}^{2}, δ = {Ar}^{3}$
$∵$ $α$ and $β$ are the roots of the equation $x^{2} - 3 x + a = 0$ ,
then
$\begin{aligned} γ + δ = 12 \\ \Rightarrow {Ar}^{2} (1 + r) = 12 . . . . (ii) \end{aligned}$
On solving Eqs. (i) and (ii), we get
$A = 1, r = 2$
$\begin{array}{ll} \Rightarrow α = 1, β = 2, γ = 4, δ = 8 \\ ∴ a = αβ = 1 \times 2 = 2 \\ and b = γδ = 4 \times 8 = 32 \end{array}$

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