The correct option is A Always passes through a fixed point
Solving equation of parabola with x-axis ( y=0)
We get (a−b)x2+(b−c)x+(c−a)=0, which should have two equal values of x, as x-axis touches the parabola ⇒(b−c)2−4(a−b)(c−a)=0
⇒(b+c−2a)2=0⇒−2a+b+c=0⇒ax+by+c=0 always passes through (-2, 1) (If we substitute (-2,1) in the equation of the line, then we get -2a+b+c=0, which is true)