Find the equation of all lines having slope $–1$ that are tangents to the curve $y= \large\frac{1}{x-1},$$\: x \neq 1$

Question

$\begin{array}{1 1} (A)\;x+y+1=0 ; x+y-3=0 \\ (B)\;x+y-1=0 ; x+y-3=0 \\ (C)\;x+y+1=0 ; x-y-3=0 \\ (D)\;x-y-1=0 ; x+y-3=0 \end{array}$