# Transformation of Axes

### Transformation of Axes:

#### Definition:

• Concept: The transformation of axes is a technique used to change the orientation or scale of the coordinate axes while preserving the shape and properties of geometric figures.

#### Types of Transformations:

1. Translation of Axes:

• Definition: Shifting the origin of the coordinate system to a new location.
• Equation: ${x}^{\mathrm{\prime }}=x-a,{y}^{\mathrm{\prime }}=y-b$ translates the origin to $\left(a,b\right)$.
2. Rotation of Axes:

• Definition: Rotating the coordinate axes by a certain angle $\theta$ about the origin.
• Equation: ${x}^{\mathrm{\prime }}=x\mathrm{cos}\theta -y\mathrm{sin}\theta ,{y}^{\mathrm{\prime }}=x\mathrm{sin}\theta +y\mathrm{cos}\theta$ rotates the axes by angle $\theta$.
3. Scaling of Axes:

• Definition: Changing the scale of the coordinate axes.
• Equation: ${x}^{\mathrm{\prime }}=kx,{y}^{\mathrm{\prime }}=ky$ scales the axes by a factor of $k$ along both axes.

#### Steps for Transformation:

1. Identify Transformation Type: Determine the type of transformation required (translation, rotation, or scaling).
2. Apply Transformation Equations: Use the appropriate transformation equations based on the type of transformation needed.
3. Apply to Coordinates: Apply these equations to the coordinates of points or equations to obtain transformed coordinates or equations.

#### Purpose:

• Simplification: Transformation of axes simplifies problem-solving by adjusting the axes to align better with the given problem.
• Analysis: Helps in analyzing geometric shapes, especially when the original axes' orientation complicates the problem.

### Example:

Consider the transformation where the original coordinate axes are translated by $4$ units to the right and $3$ units upwards. Express the equation ${x}^{2}+{y}^{2}=25$ in terms of the new coordinate system.

#### Steps to Solve:

1. Translation of Axes:

• Given the translation, the new equations for the axes are ${x}^{\mathrm{\prime }}=x-4$ and ${y}^{\mathrm{\prime }}=y-3$.
2. Apply Transformation to Equation:

• Start with the equation ${x}^{2}+{y}^{2}=25$.
• Substitute the expressions for $x$ and $y$ in terms of ${x}^{\mathrm{\prime }}$ and ${y}^{\mathrm{\prime }}$ based on the transformation of axes.
3. Transformation Equations:

• $x={x}^{\mathrm{\prime }}+4$ and $y={y}^{\mathrm{\prime }}+3$.
4. Substitution:

• Substitute the expressions for $x$ and $y$ into the equation ${x}^{2}+{y}^{2}=25$:

$\left({x}^{\mathrm{\prime }}+4{\right)}^{2}+\left({y}^{\mathrm{\prime }}+3{\right)}^{2}=25$

5. Simplify the Equation:

• Expand and simplify the equation to express it solely in terms of ${x}^{\mathrm{\prime }}$and ${y}^{\mathrm{\prime }}$.

${x}^{\mathrm{\prime }2}+8{x}^{\mathrm{\prime }}+16+{y}^{\mathrm{\prime }2}+6{y}^{\mathrm{\prime }}+9=25$

${x}^{\mathrm{\prime }2}+{y}^{\mathrm{\prime }2}+8{x}^{\mathrm{\prime }}+6{y}^{\mathrm{\prime }}=0$

#### Interpretation:

The equation of the circle ${x}^{2}+{y}^{2}=25$ in the original coordinate system, after the transformation of axes by translating $4$ units to the right and $3$ units upwards, becomes ${x}^{\mathrm{\prime }2}+{y}^{\mathrm{\prime }2}+8{x}^{\mathrm{\prime }}+6{y}^{\mathrm{\prime }}=0$ in terms of the new coordinate system. This transformation enables the representation of the circle in a modified coordinate system, simplifying its form and orientation for analysis.