# Translation of Axes

If in the plane with the given $X$ and $Y$ axes new coordinate axes are chosen parallel to the given ones, we say that there has been a translation of axes in the plane. Let $P\left( {x,y} \right)$ be any point in the $XY$-plane. Let $O’\left( {h,k} \right)$ be the fixed point in the $XY$- plane. We draw two perpendicular axes through $O’$: the $X$-axis is parallel to the $x$-axis and the $Y$-axis parallel to the $y$-axis, as shown in the given diagram. In fact, $O’$ is the origin of the new $XY$-plane. The point $P$ has the coordinates $\left( {X,Y} \right)$ with respect to the $XY$-plane.

Now
$\begin{gathered} X = O’C = AB = OB – OA = x – h\,\,\,\,\,\,\,\because X = O’C,\,\,OB = x,\,\,OA = h \\ Y = CP = BP – BC = y – k\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\because Y = CP,\,\,BP = y,\,\,BC = k \\ \end{gathered}$

The equations $X = x – h$, $Y = y – k$ are called transformation equations and are used to find the coordinates of a point with respect to the new coordinate system, the $XY$-system. Thus, the point $P\left( {x,y} \right)$ with respect to the XY-plane is $P\left( {x – h,y – k} \right)$.

Conversely, if the coordinates of a point with respect to the $XY$-system are given, then the coordinates with respect to the original system can be determined by the equations $x = X + h$, $y = Y + k$.

Example 1: Let $P\left( {8,3} \right)$ and $O’\left( {2, – 5} \right)$ be two points in the XY-coordinates system. Find the XY-coordinates of $P$ referring to the translated axes $O’X$ and $O’Y$.

Solution: Here $x = 8,\,\,y = 3$ and $h = 2,\,\,k = – 5$. The coordinates of $P$ referring to the new $XY$-coordinates system are
$\left( {x – h,y – k} \right) = \left( {8 – 2,3 – \left( { – 5} \right)} \right) = \left( {6,8} \right)$

Example 2: Let $P\left( {3,4} \right)$ be a point referring to the $XY$-coordinate system translated thorough $O’\left( {5,6} \right)$. Find the coordinates of $P$ referring to the original coordinate system,  the $xy$-system.

Solution: Here $X = 3,\,\,Y = 4$ and $h = 5,\,\,k = 6$. The coordinates of $P$ referring to the new XY-coordinates system are
$\left( {X + h,Y + k} \right) = \left( {3 + 5,4 + 6} \right) = \left( {8,10} \right)$