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Question

# Find what the following equations become when the origin is shifted to the point (1,1)?(i) x2+xy−3x−y+2=0 (ii) x2−y2−2x+2y=0(iii) xy−x−y+1=0 (iv) xy−y2−x+y=0

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Solution

## 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+kGiven origin is shifted to (1,1)∴h=1,k=1x=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=0X2+1+2X+XY+X+Y+1−3X−3−Y−1+2=0X2+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)=0X2+2X+1−Y2−1−2Y−2X−2+2Y+2=0X2−Y2=0∵X2−Y2=0, is the required equation(iii) xy−x−y+1=0(X+1)(Y+1)−(X+1)−(Y+1)+1=0XY+X+Y−X−1−Y−1+1=0XY−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)=0XY+X+Y+1−Y2−1−2Y−X−1+Y+1=0XY−Y2−Y=0∴XY−Y2−Y=0, is the required equation.

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