Q.When the origin is shifted to the point $(2, 3)$ the transformed equation of a curve is $x^2 + 3xy - 2y^2 + 17x - 7y - 11 = 0$. Find the original equation of the curve.

Translation of axes is the simplest of the two basic coordinate transformations: the origin is moved to a new point $(h,k)$, but the two axes keep their original directions — the new $x'$-axis remains parallel to the old $x$-axis and the new $y'$-axis remains parallel to the old $y$-axis. The point being described does not move at all; only the coordinates used to label it change, because the frame of reference has shifted.

If a point $P$ has coordinates $(x,y)$ with respect to the original origin $O$, and coordinates $(x',y')$ with respect to the new origin $O'(h,k)$ (measured in the original frame), then since the new axes are parallel to the old,

and solving for the new coordinates,

To transform a given equation from the old variables to the new ones, substitute $x=x'+h$ and $y=y'+k$ throughout and simplify — this is the direction used whenever a problem says the origin is shifted to $(h,k)$ and asks for the new equation of a curve. …