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Let three quadratic equations ax22bx+c=0,bx22cx+a=0 and cx22ax+b=0, all have only positive roots. Then which of these are always true?

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b2 = ac
c2 = ab
each pair of equations has exactly one root common
each pair of equations has two roots common

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detailed solution

Correct option is A

For each equation, required conditions(i) Discriminant≥0⇒b2≥ac,c2≥ab,a2≥bc(ii) f(0)>0 ⇒c,a,b>0(iii) Abscissa of vertex > 0 ⇒ba>0,cb>0,ac>0.Now, b2≥ac,c2≥ab⇒b2c2≥a2bc⇒bc≥a2But a2≥bc. So, a2=bc.Similarly, b2 = ac and c2 = ab.Therefore, a = b = c.

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