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ISRO 2013- Quadractic equation [EE]

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If the sum of the roots of the quadratic equation a$x^2$ + b$x$ + c = 0 is equal to the sum of the squares of their reciprocals, then a/c, b/a and c/b are in:
(A) arithmetic progression

(B) geometric progression

(C) arithmetic-geometric-progression

(D) harmonic progression
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Option D is Correct.

Let the roots of the equation be alpha and beta.

sum of the roots of the quadratic equation ax 2 + bx + c = 0 is equal to the sum of the squares of their reciprocals.

α + β = (1/α2 ) + (1/β2)  =  (α2 + β2) / (α*β)2 =  ((α + β)2 - 2α*β) / (α*β)2

Therefore, Put Sum of roots = - (b/a) and Product of roots is c/a  in ((α + β)2 - 2α*β) / (α*β)2

-(b/a) = ( b2 / a2 - 2*(c/a) ) / (c/a)2 ......  (Sum of roots = - (b/a) and Product of roots is c/a ....Acc to Vieta's formula for quadratic equation)

-(b/a) = ( b2 / c2 ) - 2a/c

2(a2)*c = a(b2) + b(c2) ............(Divide the equation by abc)

2a/b = b/c +c/a

c/a , a/b and b/c are in AP

So a/c, b/a and c/b are in HP.

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