The sum of the roots of the equation ax ^2+ x + c =0 (where a and c are non-zero) is equal to the sum of the reciprocals of their squares. Then a, ca ^2, c ^2 are in

Step-by-step solution

Let , be roots of a x^2+x+c=0 . Given, + = 1 ^2 + 1 ^2 = ^2+ ^2 ^2 ^2 = ( + )^2-2 ( )^2 aligned & -1 a = ( -1 a )^2-2 ( c a ) ( c a )^2 & -1 a = 1 a ^2 - 2 c a c ^2 a ^2 & -1 a = 1-2 ac c ^2 -c^2=a-2 a^2 c 2 a^2 c=a+c^2 aligned