Cartesian System of Rectangular Co-Ordinates, Shifting of Origin

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Shifting of Origin

  • We know that by drawing x-axis and y-axis, any plane is divided into four quadrants and we represent any point in the plane as an ordered pair of real numbers which are the lengths of perpendicular distances of the point from the axes drawn.

  • We also know that these axes can be chosen arbitrarily and therefore the position of these axes in the plane is not fixed.

  • Position of the axes can be changed. When we change the position of axes, the coordinates of a point also get changed correspondingly.

  • Consequently equations of curves also get changed.

The axes can be changed or transformed in the following ways:

(i) Translation of axes

(ii) Rotation of axes

(iii) Translation and rotation of axes.

In the present section we shall discuss only one transformation i.e. translation of axes.

Shifting of Origin

Shifting of Origin

Shifting of Origin

The transformation obtained, by shifting the origin to a given point in the plane, without changing the directions of coordinate axes is called translation of axes.

Let us see how coordinates of a point in a plane change under a translation of axes. Let and be the given coordinate axes. Suppose the origin is shifted to by the translation of the axes and . Let and be the new axes as shown in the above figure. Then with reference to and the point has coordinates .

Let be a point with coordinates in the system and and with coordinates in the system and . Then and .

Now

.

And .

Hence or

If the origin is shifted to by translation of axes then coordinates of the point are transformed to and the equation of the curve is transformed to .

Translation formula always hold, irrespective of the quadrant in which the origin of the new system happens to lie.

Example 1:

When the origin is shifted to by translation of axes find the coordinates of the point with respect to new axes.

Solution:

Here

Therefore

Example 2:

When the origin is shifted to the point by the translation of axes, find the transformed equation of the line

Solution:

Here

and .

Substituting the values of and in the equation of line

We get

i.e. .

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