(i)  I read /(edit) this condition in this way:

         One root is of opposite sign to the other root.


     Then, according to the Vieta's theorem, the coefficient "b" at "x" is 

         b =  = 0,  since the sum of the roots is zero.


     Thus, the necessary and sufficient condition for the equation to have two real roots of opposite signs is

         a) the coefficient b= 0,     and
         b) the coefficients  "a"  and  "c"  are of opposite signs.


     If complex roots are allowed, then the single condition a) is necessary and sufficient.



(iii)  If the roots are reciprocal each other, then their product is equal to 1.

       From the other side, the product of the roots is equal to  , according to Vieta's theorem.

       Therefore, the necessary and sufficient condition for the given equation to have reciprocal roots is   = 1,   or  simply c = a.