If 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

Let a variable point on the parabola be $\left(1+t^2, 2t\right)$ and its reflection in the given line be (h, k)

$h-(1+t{)}^2=k-2t=\frac{-2\left(1+t^2+2t-2\right)}{2}$

$h-(1+t{)}^2=-1-t^2-2t+2\Rightarrow h=2(1-t)$ and

$k=-\left(t^2-1\right)$

$\frac{k}{h}=\frac{-(t-1)(t+1)}{2(1-t)} \therefore t+1=\frac{2k}{h}$ and $t=\frac{2k}{h}-1$

$1-t=2-\frac{2k}{h}=\frac{k}{2}$

$2h-2k=\frac{h}{2}$

Locus of (h, k) is

$4x-4y=x^2$

$A+B=0$