Translation of Axes [Definition and Meaning] -

Translation of Axes: Let $xy-$ coordinate system be given and $O'( h,k)$ be any point in the plane. Through $O'$ draw two mutually perpendicular lines $O'X, O'Y$ such that $O'X$ is parallel to $Ox$ . The new axes $O'X$ and $O'Y$ are called translation of the $Ox$ and $Oy$ axes through the point $O'$ . In this, the origin is shifted to another point in the plane but the axis remains parallel to the old axes. Let $P$ be a point with coordinates $(x,y )$ referred to $xy$ -coordinate system and the axes be translated through the point $O'(h ,k )$ and $O'X, O'Y$ be the new axes. If $P$ has coordinates $(X, Y)$ referred to the new axes, then we need to find $X, Y$ in terms of $x, y$ .

Translation of Axes

Draw $PM$ and $O' N$ perpendiculars to $Ox$ .

From the figure, we have

$OM =x,\;\;\;\;\;MP= y,\;\;\;\;\;ON= h,\;\;\;\;\;NO'= k =MM'$ .

Now

$X =O'M' =NM= OM -OM- ON =x -h$ .

Similarly,

$Y =M' P =MP- MM'= y -k$

Thus the coordinates of $P$ referred to $XY$ -system are $(x-h,y-k )$ .

$i.e,$

$X= x-h$

$Y= y-k$ .

Moreover,

$\boxed{x= X+ h \;\;\;\;and\;\;\;\; y=Y+k}$

Example 1:(Translation of Axes)

The coordinates of a point P are $(-6, 9)$ . The axes are translated through the point $O' (-3, 2)$ . Find the coordinates of Preferred to the new axes.
Solution:

Here

$h=-3\;\;\; and\;\;\;\; k= 2$

Coordinates of P referred to the new axes are (X, Y) given by

$X =x-h= -6 - (-3) = -3$

And

$Y =y-k= 9 - 2 = 7$

Thus

$\boxed{P (X, Y) = P (-3 ,7)}$ .

Example 2: (Translation of Axes)

The $xy$ -coordinate axes are translated through the point $O' (4, 6)$ . The coordinates of the point P are $(2, -3)$ referred to as the new axes. Find the coordinates of Preferred to the original axes.
Solution: