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=xa,y=yb translates the origin to (a,b).
  2. Rotation of Axes:

    • Definition: Rotating the coordinate axes by a certain angle θ about the origin.
    • Equation: x=xcosθysinθ,y=xsinθ+ycosθ rotates the axes by angle θ.
  3. Scaling of Axes:

    • Definition: Changing the scale of the coordinate axes.
    • Equation: x=kx,y=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 x2+y2=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=x4 and y=y3.
  2. Apply Transformation to Equation:

    • Start with the equation x2+y2=25.
    • Substitute the expressions for x and y in terms of x and y based on the transformation of axes.
  3. Transformation Equations:

    • x=x+4 and y=y+3.
  4. Substitution:

    • Substitute the expressions for x and y into the equation x2+y2=25:

    (x+4)2+(y+3)2=25

  5. Simplify the Equation:

    • Expand and simplify the equation to express it solely in terms of xand y.

    x2+8x+16+y2+6y+9=25

     x2+y2+8x+6y=0

Interpretation:

The equation of the circle x2+y2=25 in the original coordinate system, after the transformation of axes by translating 4 units to the right and 3 units upwards, becomes x2+y2+8x+6y=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.