12.1 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 axis parallel to the originals that can enable working with equations more simply.

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

e12-2.jpg

The new origin O’ is defined and the axis x’ and y’ are parallel to the original axis 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:

explica12_clip_image002.gif  Solving for x’ and y’, we obtain the new coordinates of the point: explica12_clip_image004.gif

Example

The origin be the point being O’(-2,5), find the transformed equation of the curve:

explica12_clip_image006.gif

Steps Procedure

Identify the values for h and k, which are the coordinates of the new origin.

explica12_clip_image008.gif

In the equation to transform, substitute the previous relations:

explica12_clip_image012.gif

By expanding the previous expression and simplifying:

explica12_clip_image014.gif

Answer

The simplified equation is:

explica12_clip_image016.gif 

12.2 Rotation of axis

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

Rotation of axis is the process of making the transformation of coordinates of a system of axis x and y to another of axis x’ and y’, by the rotation of axis in angle explica12_clip_image018.gif.

e12-3.jpg

The equations for rotation in their trigonometric form are:

explica12_clip_image020.gif

Example

Transform the equation explica12_clip_image022.gif by turning the axis in an angle of 30º.

Steps Procedure

Substitute the value of the angle in the equations of rotation:

explica12_clip_image024.gif

Substitute the value of the trigonometric functions:

We have that:

explica12_clip_image026.gif      Then:       explica12_clip_image028.gif

Substitute the values of
x and y in the equation:

explica12_clip_image030.gif

Expanding:

explica12_clip_image032.gif
By multiplying the whole equation by 4 to eliminate denominators, we have:

explica12_clip_image034.gif

We can divide the whole equation by 2 to simplify the coefficients.

Answer

The transformed equation is:

explica12_clip_image036.gif

When the angle of rotation of the axis is not known and given the general equation of second degreeexplica12_clip_image038.gif, the following formula should be used:

explica12_clip_image040.gif

Example

Simplify the following equation by the rotation of axis to eliminate the term Bxy:

explica12_clip_image042.gif

Steps Procedure

Obtain the angle of rotation of the curve:

explica12_clip_image044.gif          

Calculate the equations of rotation:

explica12_clip_image048.gif

Substitute the equations of rotation in the equation to simplify:

explica12_clip_image052.gif

Expanding:

explica12_clip_image054.gif

Expanding (continued):

explica12_clip_image056.gif

Simplifying:

explica12_clip_image058.gif

Answer

The simplified equation is:

explica12_clip_image060.gif