Q.When the origin is shifted to $(-1,2)$ by translation of axes, find the transformed equation of $x^2+y^2+2x-4y+1=0$.

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.

Translation is the natural tool for simplifying equations of curves such as circles, parabolas, ellipses and hyperbolas whose centre or vertex is not at the origin but whose axes are still parallel to the coordinate axes. Shifting the origin to that centre or vertex removes the linear ($x$ and $y$) terms from the equation, leaving only the squared terms and a constant — exactly what the standard form of such a curve requires. Translation alone, however, can never remove a genuine $xy$ term, since substituting $x=x'+h,\,y=y'+k$ into a term like $2hxy$ only ever produces new linear and constant contributions, not a change to the coefficient of $x'y'$ itself. Removing an $xy$ term needs a rotation of axes instead.