Question

If the sum of the roots of the quadratic equation ax2 + bx + c = 0 is equal to the sum of the squares of their reciprocals, then a/c, b/a, c/b are in

Answer 1

let A,B  be the roots of this equation then,


  A+B=-b/a  & AB=c/a          .................1


now it is given that sum of roots is equal to square of resiprocal of roots


so


         A+B=1/A2 +1/B2


        A+B=A2+B2 /(AB)2 


        A+B   = [(A+B)2 -2AB]/(AB)2


now putting value of A+B and AB from eq 1


      -b/a =  [b2/a2 -2c/a] /c2/a2


      ab2 +bc2 =2a2c                      ........................2


now let a/c,b/a,c/b are in HP then


           a/b = [b/c + c/a]/2


        ab2 + bc2 =2a2c


this result is similar to eq 2 so these terms are in HP.........