Question

By shifting origin to a suitable point with out rotation of axes, the equation $xy-x+2y=6$ has transformed to $XY=B$, then the value of $B$ is

Answer 1

Let the new origin be $(h,k)$
$\therefore x=X+h,y=Y+k$
Original equation: $xy-x+2y=6$

$\Rightarrow (X+h)(Y+k)-(X+h)+2(Y+k)=6$
$\Rightarrow XY+kX+hY+hk-X-h+2Y+2k=6$

$\Rightarrow XY+(k-1)X+(h+2)Y=6-hk+h-2k$
Comparing with $XY=B$, we get

$k-1=0,h+2=0\text{and}B=6-hk+h-2k$

$\Rightarrow k=1,h=-2$

$\therefore B=6+2-2-2=4$