Question

$\text{Find what the following equations become when the origin is shifted to the point(1,1)}?\left(i\right)x^2+xy-3y-y+2=0\left(ii\right)x^2-y^2-2x+2y=0\left(iii\right)xy-x-y+1=0\left(iv\right)xy-y^2-x+y=0$

Answer 1

$Wehave,\left(i\right)x^2+xy-3y-y+2=0substitutingx=X+1,Y+1intheequation,weget:(X+1{)}^2+(X+1\left)\right(Y+1)-3(X+1)-(Y+1)+2=0\Rightarrow X^2+2X+2+XY+X+Y+1-3X-3-Y-1+2=0\Rightarrow X^2+XY=0Hecne,thetransformedequatinis{x}^2+xy=0(ii)x^2-y^2-2x+2y=0substitutingx=X+1,Y+1intheequation,weget:(X+1{)}^2-(Y+1{)}^2-2(X+1)+2(Y+1)=0\Rightarrow X^2+1+2X-(Y^2+1+2Y)-2X-2+2Y+2=0\Rightarrow X^2+1-Y^2-1-2Y+2Y=0\Rightarrow X^2-Y^2=0Hencethetransformedequationis{x}^2-y^2=0(iii)xy-x-y+1=0substitutingx=X+1,Y+1intheequation,weget:(X+1\left)\right(Y+1)-(X+1)-(Y+1)+1=0\Rightarrow XY+X+Y+1-X-1-Y-1+1-0\Rightarrow XY+X+Y+1-X-1-Y-1+1=0\Rightarrow XY=0Hencethetransformedequationisxy=0(iv)xy-y^2-x+y=0substitutingx=X+1,Y+1intheequation,weget:(X+1\left)\right(Y+1)-(Y+1)-(X+1)+(Y+1)=0\Rightarrow XY+X+Y+Y+1-(Y^2+1+2Y)-X-1+Y+1=0\Rightarrow XY+2Y-Y^2-1-2Y+1=0\Rightarrow XY-Y^2=0Hencethetransfermedequationisxy-y^2=0$