If - - are the roots of a x2-b x-c-0y-0 and - -2-2- -3-3 are in G-P - then the value of c - is where - b 2-4 ac

Ax^2+bx+c = 0 nbsp; ⇒αβ=ca α+β, α^2 +β^2 , α^3 +β^3 are in G.P. ⇒(α+β) × (α^3 +β^3) = (α^2 +β^2)^2 ⇒β^3 α +βα^3 = 2α^2β^2 ⇒β^3 α α^2β^2 α^2β^2 +βα^3 = 0 ⇒β ...

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Byju's Answer

Standard XII

Mathematics

Geometric Progression

If α, β are t...

Question

If $α, β$ are the roots of $a x^{2} + b x + c = 0 (a \neq 0)$ and $α + β, α^{2} + β^{2}, α^{3} + β^{3}$ are in $G . P .$ , then the value of $c \cdot Δ$ is
(where $Δ = b^{2} - 4 a c$ )

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Solution

$a x^{2} + b x + c = 0$
$\Rightarrow α β = \frac{c}{a}$

$α + β, α^{2} + β^{2}, α^{3} + β^{3}$ are in $G . P .$
$\Rightarrow (α + β) \times (α^{3} + β^{3}) = (α^{2} + β^{2})^{2} \Rightarrow β^{3} α + β α^{3} = 2 α^{2} β^{2} \Rightarrow β^{3} α - α^{2} β^{2} - α^{2} β^{2} + β α^{3} = 0 \Rightarrow β^{2} α (β - α) - β α^{2} (β - α) = 0 \Rightarrow (β - α) (β α^{2} - α β^{2}) = 0 \Rightarrow α β (α - β)^{2} = 0 \Rightarrow α β = 0 or α = β \Rightarrow c = 0 or Δ = 0 ∴ c \cdot Δ = 0$

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If α,β are the roots of ax2+bx+c=0 (a≠0) and α+β,α2+β2,α3+β3 are in G.P., then the value of c⋅Δ is (where Δ=b2−4ac)

ax2+bx+c=0 ⇒αβ=ca α+β,α2+β2,α3+β3 are in G.P. ⇒(α+β)×(α3+β3)=(α2+β2)2⇒β3α+βα3=2α2β2⇒β3α−α2β2−α2β2+βα3=0⇒β2α(β−α)−βα2(β−α)=0⇒(β−α)(βα2−αβ2)=0⇒αβ(α−β)2=0⇒αβ=0 or α=β⇒c=0 or Δ=0∴c⋅Δ=0

If $α, β$ are the roots of $a x^{2} + b x + c = 0 (a \neq 0)$ and $α + β, α^{2} + β^{2}, α^{3} + β^{3}$ are in $G . P .$ , then the value of $c \cdot Δ$ is
(where $Δ = b^{2} - 4 a c$ )