Relation between Roots and Coefficients for Quadratic
Let p and q b...
Question
Let p and q be real numbers such that p≠0, p3≠2q and p3≠−q. If α and β are non zero complex numbers satisfying α+β=−p and α3+β3=q, then a quadratic equation having αβ and βα as its roots is
A
(p3+q)x2−(p3+2q)x+(p3+q)=0
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B
(p3+q)x2−(p3−2q)x+(p3+q)=0
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C
(p3−q)x2−(p3−2q)x+(p3+q)=0
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D
(p3−q)x2−(p3+2q)x+(p3−q)=0
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Solution
The correct option is B(p3+q)x2−(p3−2q)x+(p3+q)=0 Given that α+β=−pandα3+β3=q⇒(α+β)3−3αβ(α+β)=q⇒−p3−3αβ(−p)=q⇒αβ=p3+q3p Now for required quadratic equation, sum of roots =αβ+βα=α2+β2αβ=(α+β)2−2αβαβ=p2−2(p3+q3p)p3+q3p=3p3−2p3−2qp3+q=p3−2qp3+q and product of roots =αβ.βα=1 ∴ Required equation is x2−(p3−2qp3+q)x+1or(p3+q)x2−(p3−2q)x+(p3+q)=0