Step-by-step Solution:

Equation can be written as \((x-y)^{2}=4(x+y-1)\) Suppose \((x-y)=Y\) and \((x+y-1)=X\), \[\] then equation becomes \[\] \(Y^{2}=4 X\), whose focus will be \((1,0)\). \[\] Thus \(X=1\) and \(Y=0\) \[\] Or \(x+y-1=1\) and \(x-y=0\) \[\] \(\Rightarrow x=1, y=1\) \[\] Choice (A)\[\]