the sum of the root of the equation x2+bx+c where b and c are non zero is equal to sum of reciprocal of their square the 1/c,b,c/b are in which series

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

-(b) = ( b2 / c2 ) - 2/c

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

2/b = b/c +c

c/a , a/b and b/c are in AP

So 1/c, b and c/b are in HP.