In the quadratic equation ax2+bx+c=0,Δ=b2−4acandα+β,α2+β2,α3+β3, are in G.P. where α,β are the root of ax2+bx+c=0, then
A
Δ≠0
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B
bΔ=0
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C
cΔ=0
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D
Δ=0
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Solution
The correct option is CcΔ=0 In the quadratic equation ax2+bx+c=0 Δ=b2−4acandα+β=−ba.αβ=caα2+β2=(α+β)2−2αβ=b2a2−2ca=b2−2aca2andα3+β3=−b3a3−3ca(−ba)=−(b3−3abca3)Givenα+β,α2+β2,α3+β3areinG.P.⇒−ba,b2−2aca2,−(b3−3abc)a3areinG.P.⇒(b2−2aca2)2=ba(b3−3abca3)⇒b4+4a2c2−4ab2c=b4−3ab2c⇒4a2c2−ab2c=0⇒acΔ=0⇒cΔ=0(∵Inquadratic a ≠0)