Axis Translation and Rotation: Simplifying Conic Equations

Written at March 23, 2025 on English with a size of 9.41 KB.

Translation of Axis

When you need to simplify equations, mainly the conics (circumference, parabola, ellipse, and hyperbola), you need to create a pair of coordinate axes parallel to the originals. This enables working with equations more simply.

This creation of axes is known as parallel translation of axes and consists of moving one or both axes so that the origin moves to a new position, and the axes are parallel to the original.

In the following figure, you can see how you can translate the equations of the curves from one Cartesian plane x and y to a Cartesian plane x’ and y’.

The new origin O’ is defined, and the axes x’ and y’ are parallel to the original axes x and y, so that for a point P(x,y) you have P(x’,y’). With this, the coordinates of the point in the original system are given by:

Solving for x’ and y’, we obtain the new coordinates of the point:

Example
The origin is the point O’(-2,5). Find the transformed equation of the curve:
Steps	Procedure
Identify the values for h and k, which are the coordinates of the new origin.
In the equation to transform, substitute the previous relations:
By expanding the previous expression and simplifying:
Answer
The simplified equation is:

Rotation of Axis

In the previous subtopic, you studied the procedure to simplify the equations by the translation of axes. Now, the simplification will be made by the rotation of axes to eliminate the term Bxy from the equation of the curve.

Rotation of axes is the process of making the transformation of coordinates of a system of axes x and y to another of axes x’ and y’, by the rotation of axes in angle

The equations for rotation in their trigonometric form are:

Example
Transform the equation by turning the axis in an angle of 30º.
Steps	Procedure
Substitute the value of the angle in the equations of rotation:
Substitute the value of the trigonometric functions:	We have that: Then:
Substitute the values of x and y in the equation:
Expanding:	By multiplying the whole equation by 4 to eliminate denominators, we have: We can divide the whole equation by 2 to simplify the coefficients.
Answer
The transformed equation is:

When the angle of rotation of the axis is not known and given the general equation of second degree

, the following formula should be used:

Example
Simplify the following equation by the rotation of axes to eliminate the term *Bxy*:
Steps	Procedure
Obtain the angle of rotation of the curve:
Calculate the equations of rotation:
Substitute the equations of rotation in the equation to simplify:
Expanding:
Expanding (continued):
Simplifying:
Answer
The simplified equation is: