If the roots of the equation  1x+a+1x+b=1c                         are equal in magnitude but opposite in sign, then their product is

The equation (1) can be written asc(x+b+x+a)=(x+a)(x+b)or x2+(a+b−2c)x+ab−ac−bc=0                    (2)Let α and -α be the roots of (2) then0=α+(−α)=a+b−2c⇒c=12(a+b)Also, 2α(−α)=2ab−2(a+b)c=2ab−(a+b)2=−a2+b2⇒ α(−α)=−12a2+b2