Translation of Axes

If in the plane with 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.


translation-axes

Let P\left( {x,y} \right) be any point in XY-plane. Let O'\left( {h,k} \right) be the fixed point in the XY- plane. We draw two perpendicular axes through O', X-axis is parallel to x-axis and Y-axis parallel to 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 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, XY-system. Thus, the point P\left( {x,y} \right) with respect to XY-plane is P\left( {x - h,y - k} \right).
Conversely, if the coordinates of a point with respect XY-system are given, then its 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 xy-coordinates system. Find the XY-coordinates of P referred to the translated axes O'X and O'Y.

Solution: Here x = 8,\,\,y = 3 and h = 2,\,\,k =   - 5. The coordinates of P referred to 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 referred to XY-coordinate system translated thorough O'\left( {5,6} \right). Find the coordinates of P referred to the original coordinate system, xy-system.

Solution: Here X = 3,\,\,Y = 4 and h = 5,\,\,k = 6. The coordinates of P referred to new XY-coordinates system are

\left(  {X + h,Y + k} \right) = \left( {3 + 5,4 + 6} \right) = \left( {8,10} \right)

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