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Question

Find what the following equations become when the origin is shifted to the point (1,1)?
(i) x2+xy3xy+2=0 (ii) x2y22x+2y=0
(iii) xyxy+1=0 (iv) xyy2x+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+k
Given origin is shifted to (1,1)
h=1,k=1
x=X+1,y=Y+1
(i) x2+xy3xy+2=0
(X+1)2+(X+1)(Y+1)3(X+1)(Y+1)+2=0
X2+1+2X+XY+X+Y+13X3Y1+2=0
X2+XY=0
X2+XY=0, is the required equation
(ii) x2y22x+2y=0
(X+1)2(Y1)22(X+1)+2(Y+1)=0
X2+2X+1Y212Y2X2+2Y+2=0
X2Y2=0
X2Y2=0, is the required equation
(iii) xyxy+1=0
(X+1)(Y+1)(X+1)(Y+1)+1=0
XY+X+YX1Y1+1=0
XY1=0
XY1=0, is the required equation
(iv) xyy2x+y=0
(X+1)(Y+1)(Y+1)2(X+1)+(Y+1)=0
XY+X+Y+1Y212YX1+Y+1=0
XYY2Y=0
XYY2Y=0, is the required equation.

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