Q : The quadratic equation whose roots are A.M. and H.M. between the roots of the equation ax^{2} + bx + c = 0 is

(A) abx^{2} + (b^{2} + ac)x + bc = 0

(B) 2abx^{2} + (b^{2} + 4ac)x + 2bc = 0

(C) 2abx^{2} + (b^{2} + ac)x + bc = 0

(D) none of these

Ans: (B)

Solution: Let (α , β) be the roots of the given equation, then

$\large \alpha + \beta = -\frac{b}{a} \; , \alpha . \beta = \frac{c}{a} $

Required equation is

$\large x^2 – (\frac{\alpha + \beta}{2} + \frac{2 \alpha . \beta}{\alpha + \beta})x + (\frac{\alpha + \beta}{2}) (\frac{2 \alpha . \beta}{\alpha + \beta})) = 0 $

2abx^{2} + (b^{2} + 4ac)x + 2bc = 0

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