reflection of a parabola about a line

Question

If the reflection of the parabola $y^2$ = $4(x – 1)$ in the line $ x + y = 2$ is the curve $Ax + By = x^2$, then the value of $(A + B)$ is $?$

My try: I know one of the method is to take the reflection of the co-ordinates of vertex and focus of the parabola and then proceeding further. Isn't there any simple way to find the reflected curve ?

Answer 1

The invariant points under the transformation are the intersections of each curve with the line $x+y=2$

So solving simultaneously, the intersections satisfy both quadratic equations $$x^2=8x-8$$ and $$x^2=x(A-B)+2B$$

We therefore have $$A-B=8$$ and $$2B=-8$$ from which we obtain $B=-4$ and $A=4$ so $$A+B=0$$

Answer 2

From given line of reflection

$$ x= 2 -y_1,\, y= 2-x_1\, $$

Plug in and simplify

$$ 4 x_1 -4 y_1= x_1^2 ==A x_1 - B y_1 = x^2; \, A+B=0. $$