The sum of the roots of the equation x^2+b x+c=0 (where b and c are non-zero) is equal to the sum of the reciprocals of their squares. Then 1 c , b , c b are in

Step-by-step solution

Let , be the roots of x^2+ bx + c =0 Given, + = 1 ^2 + 1 ^2 aligned & + =- b ; = c & + = 1 ^2 + 1 ^2 & + = ^2 ^2 + ^2 ^2 = ( + )^2-2 ^2 ^2 aligned aligned & - b = b ^2-2 c c ^2 & - bc ^2= b ^2-2 c 2 c = b ^2+ bc ^2 & 2 c ~b = b + c ^2 2 ~b = b c + c aligned We know, if a , b , c are in A.P. 2 ~b = a + c . Similarly, we got 2 ~b = b c + c , which means b c , 1 ~b , c are in A.P. So, c b , b , 1 c are in H.P.