Let (x,y) be the old co-ordinates and (X,Y) be the new coordinates after shifting the origin to (h,k)
Then x=X+h, y=Y+k
Given origin is shifted to (1,1)
∴h=1,k=1
x=X+1,y=Y+1
(i) x2+xy−3x−y+2=0
(X+1)2+(X+1)(Y+1)−3(X+1)−(Y+1)+2=0
X2+1+2X+XY+X+Y+1−3X−3−Y−1+2=0
X2+XY=0
∴X2+XY=0, is the required equation
(ii) x2−y2−2x+2y=0
(X+1)2−(Y−1)2−2(X+1)+2(Y+1)=0
X2+2X+1−Y2−1−2Y−2X−2+2Y+2=0
X2−Y2=0
∵X2−Y2=0, is the required equation
(iii) xy−x−y+1=0
(X+1)(Y+1)−(X+1)−(Y+1)+1=0
XY+X+Y−X−1−Y−1+1=0
XY−1=0
∴XY−1=0, is the required equation
(iv) xy−y2−x+y=0
(X+1)(Y+1)−(Y+1)2−(X+1)+(Y+1)=0
XY+X+Y+1−Y2−1−2Y−X−1+Y+1=0
XY−Y2−Y=0
∴XY−Y2−Y=0, is the required equation.