Translation of Axis

In coordinate geometry, the translation of axes is a method used to simplify the equation of a curve by shifting the origin to a new point. This technique is particularly useful when dealing with equations of conics or straight lines that do not have their center at the origin. By translating the axes, we can often convert the equation of a curve into a standard form, which is easier to analyze and understand.

Understanding Translation of Axis

When we talk about translating the axis, we are referring to moving the coordinate system's origin to a new point without rotating the axes. This means that the new axes are parallel to the original axes.

Let's denote the original coordinate system as $(x, y)$ and the translated coordinate system as $(X, Y)$. If we translate the origin to a new point $(h, k)$, the relationship between the old and new coordinates is given by:

[ \begin{align*} X &= x - h \ Y &= y - k \end{align*} ]

Conversely, to go back to the original coordinates from the translated ones, we use:

[ \begin{align*} x &= X + h \ y &= Y + k \end{align*} ]

Differences and Important Points

Here is a table summarizing the key differences and important points of the original and translated axes:

Aspect	Original Axes $(x, y)$	Translated Axes $(X, Y)$
Origin	$(0, 0)$	$(h, k)$
Relation to Old Axes	N/A	$X = x - h$, $Y = y - k$
Relation to New Axes	$x = X + h$, $y = Y + k$	N/A
Equation Transformation	More complex	Simplified

Formulas in Translation of Axis

When translating the axis, the equations of geometric figures also change. For example, the equation of a circle with center $(a, b)$ and radius $r$ is:

[ (x - a)^2 + (y - b)^2 = r^2 ]

If we translate the axes such that the center of the circle becomes the new origin $(0, 0)$, the equation becomes:

[ (X - (a - h))^2 + (Y - (b - k))^2 = r^2 ]

Simplifying this, we get:

[ X^2 + Y^2 = r^2 ]

Examples

Example 1: Translation of a Line

Consider the line with the equation $2x + 3y - 6 = 0$. Translate the axes such that the new origin is at $(1, 2)$.

Solution:

Using the translation formulas, we have:

[ \begin{align*} X &= x - 1 \ Y &= y - 2 \end{align*} ]

Substituting $x = X + 1$ and $y = Y + 2$ into the original equation, we get:

[ 2(X + 1) + 3(Y + 2) - 6 = 0 \ 2X + 2 + 3Y + 6 - 6 = 0 \ 2X + 3Y + 2 = 0 ]

The translated equation of the line is $2X + 3Y + 2 = 0$.

Example 2: Translation of a Parabola

Consider the parabola with the equation $y^2 = 4x$. Translate the axes such that the new origin is at $(2, 0)$.

Solution:

Using the translation formulas, we have:

[ \begin{align*} X &= x - 2 \ Y &= y \end{align*} ]

Substituting $x = X + 2$ into the original equation, we get:

[ Y^2 = 4(X + 2) \ Y^2 = 4X + 8 ]

The translated equation of the parabola is $Y^2 = 4X + 8$.

By translating the axes, we can often simplify the equations of curves, making it easier to work with them in various applications, such as solving problems or graphing. It is a powerful technique in coordinate geometry that can greatly aid in the analysis of geometric figures.