Question

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

Answer 1

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 $(1,1)$

$\therefore h=1,k=1$

$x=X+1,y=Y+1$

(i) $x^2+xy-3x-y+2=0$

$(X+1{)}^2+(X+1\left)\right(Y+1)-3(X+1)-(Y+1)+2=0$

$X^2+1+2X+XY+X+Y+1-3X-3-Y-1+2=0$

$X^2+XY=0$

$\therefore X^2+XY=0,$ is the required equation

(ii) $x^2-y^2-2x+2y=0$

$(X+1{)}^2-(Y-1{)}^2-2(X+1)+2(Y+1)=0$

$X^2+2X+1-Y^2-1-2Y-2X-2+2Y+2=0$

$X^2-Y^2=0$

$\because X^2-Y^2=0,$ is the required equation

(iii) $xy-x-y+1=0$

$(X+1)(Y+1)-(X+1)-(Y+1)+1=0$

$XY+X+Y-X-1-Y-1+1=0$

$XY-1=0$

$\therefore XY-1=0,$ is the required equation

(iv) $xy-y^2-x+y=0$

$(X+1)(Y+1)-(Y+1{)}^2-(X+1)+(Y+1)=0$

$XY+X+Y+1-Y^2-1-2Y-X-1+Y+1=0$

$XY-Y^2-Y=0$

$\therefore XY-Y^2-Y=0,$ is the required equation.