If a, b, and c are in geometric progression and the roots of the equation ax2 + 2bx + c = 0 are α , β and those of cx2 + 2bx + a = 0 are γ, δ, then

1) α ≠ β ≠ γ ≠ δ

2) α ≠ β and γ ≠ δ

3) aα = aβ = cγ = cδ

4) α = β and γ ≠ δ

5) α ≠ β and γ – δ

Solution: (3) aα = aβ = cγ = cδ

Here a = a, b = aγ, c = aγ2, γ = common ratio

ax2 + 2bx + c = 0

α + β = – 2b / a = – 2aγ / a = – 2γ

α β = c / a = aγ2 / a = γ2

cx2 + 2bx + a = 0

γ + δ = – 2b / c = – 2ar / aγ2 = – 2 / γ

γ δ = a / c = a / aγ = 1 / γ2

(α – β)2 = (α + β)2 – 4 α β

(γ – δ)2 = (γ + δ)2 – 4 γ δ

α = β and γ = δ

aα = – aγ, aβ = – aγ

cγ = – aγ2 / γ = – aγ

cγ = – aγ2 / γ = – aγ

aα = aβ = cγ = cδ

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