MTH120 College Algebra Chapter 8.4

 8.4 Rotation of Axes

I'm sure it sounds like a viking axe throwing event but we are talking about a mathematical technique used to simplify the equations of conic sections (such as ellipses, hyperbolas, and parabolas) by changing the coordinate system through a rotation. This rotation transforms the standard coordinate axes (x and y) into new axes (u and v), making it easier to work with and analyze the conic section.

Here's an overview of the rotation of axes and its key concepts:

1. Standard Coordinate System: In the standard coordinate system, the x-axis and y-axis are aligned horizontally and vertically, respectively. The general equations of conic sections in the standard form can be complex.

2. Rotation Angle (θ): To simplify these equations, you can introduce a new coordinate system (u, v) by rotating the standard system by an angle θ. This new system is often chosen such that one of the axes aligns with the major or minor axis of the conic section, making the equations simpler.

3. Rotation Equations: The rotation of axes is performed using the following transformation equations:

   $�=�\cos (�)-�\sin (�)$ $�=�\sin (�)+�\cos (�)$

   Here, (x, y) are the coordinates in the original system, and (u, v) are the coordinates in the rotated system.

4. Simplified Equations: In the rotated system, the equations of conic sections often simplify. For example, the equation of an ellipse or a hyperbola might become a more standard form with no cross terms (xy).

5. Analysis and Graphing: In the rotated coordinate system, you can analyze and graph the conic section more easily because the terms involving the cross product xy disappear, and the equations become standard forms.

Rotation of axes is particularly useful when dealing with complex conic sections or systems of equations involving conic sections, as it simplifies the mathematics and allows for a clearer understanding of the shapes and orientations of these curves.

Non Degenerate conics are the conic sections that are not degenerate or trivial cases. In general, non degenerate conics in general form can be categorized into three types: ellipses, hyperbolas, and parabolas. Degenerate cases occur when a conic section becomes a single point, a line, or a pair of intersecting lines. Here are the general forms of nondegenerate conics and examples of each type:

1. Ellipse (Nondegenerate Form):

The general equation of an ellipse is:

$�{�}^2+�{�}^2+��+��+�=0$

where A, B, C, D, and E are constants, and A and B have different signs (A and B are not equal).

Example of an Ellipse:

The equation $2{�}^2+5{�}^2-6�-10�+9=0$ represents an ellipse. To determine its standard form, you would need to complete the square for both the x and y terms.

2. Hyperbola (Nondegenerate Form):

The general equation of a hyperbola is:

$�{�}^2-�{�}^2+��+��+�=0$

where A and B have opposite signs (A is positive and B is negative).

Example of a Hyperbola:

The equation $3{�}^2-4{�}^2-6�+8�+7=0$ represents a hyperbola. To find its standard form, you would complete the square for both the x and y terms.

3. Parabola (Nondegenerate Form):

The general equation of a parabola is:

$�{�}^2=��+�{�}^2+��+�$

where either A or B is non-zero, but not both.

Example of a Parabola:

The equation $3{�}^2=4�+2{�}^2-5�+6$ represents a parabola. To find its standard form, you would simplify the equation to isolate y.

It's important to note that when you encounter a general form equation of a conic section, you may need to manipulate the equation to express it in standard form for better analysis and graphing. The standard forms for ellipses, hyperbolas, and parabolas are more convenient for understanding the properties of these curves.

The general form of a conic section equation depends on the type of conic section (ellipse, hyperbola, parabola) and its orientation (vertical or horizontal). Here are the general forms for each type:

1. General Form of an Ellipse:

The general form of an ellipse equation is as follows:

$�{�}^2+�{�}^2+��+��+�=0$

In this equation:

• A and B are constants with different signs (A and B have opposite signs).
• C, D, and E are constants.

The sign of A and B determines the orientation of the ellipse. If A is positive and B is negative, it's a vertically oriented ellipse; if A is negative and B is positive, it's horizontally oriented.

2. General Form of a Hyperbola:

The general form of a hyperbola equation is as follows:

$�{�}^2-�{�}^2+��+��+�=0$

In this equation:

• A and B are constants with opposite signs (A is positive, and B is negative).
• C, D, and E are constants.

The sign of A and B determines the orientation of the hyperbola. If A is positive and B is negative, it's a vertically oriented hyperbola; if A is negative and B is positive, it's horizontally oriented.

3. General Form of a Parabola:

The general form of a parabola equation is as follows:

$�{�}^2=��+�{�}^2+��+�$

In this equation:

• A and B are constants. One of them must be non-zero, but not both.
• C, D, and E are constants.

The parabola's orientation (vertical or horizontal) depends on whether the equation involves x or y as the squared variable.

These general forms are useful for expressing the equations of conic sections in a non-standard way. However, it's often more convenient to work with conic section equations in their standard forms, as they provide a clearer understanding of the properties and orientation of the curves. Converting from general form to standard form typically involves algebraic manipulations to complete the square or otherwise simplify the equation.

Rotating a conic section equation through a given angle involves changing the coordinate system to align with the major and minor axes of the conic section. This transformation simplifies the equation. The steps for finding a new representation after rotating an equation are as follows:

1. Identify the type of conic section (ellipse, hyperbola, or parabola) and its orientation (vertical or horizontal) in the standard form equation.

2. Determine the angle of rotation (θ) needed to align the coordinate axes with the major and minor axes of the conic section. For example, for a hyperbola, θ would be such that the cross terms are eliminated, making the equation easier to work with.

3. Apply a rotation transformation to the coordinates (x, y) to find the new coordinates (u, v) that will align the axes.

   The transformation equations for rotating counterclockwise by an angle θ are as follows:

   $�=�\cos (�)-�\sin (�)$ $�=�\sin (�)+�\cos (�)$

4. Substitute the new coordinates (u, v) into the standard form equation of the conic section.

5. Simplify and rewrite the equation in terms of (u, v) to find the new representation.

Here are examples of how to apply this process to rotate a given equation through a specified angle:

Example 1: Rotating an Ellipse

Given the equation of a vertically oriented ellipse: $4{�}^2+9{�}^2=36$

To eliminate the cross terms and align the axes with the major and minor axes, we rotate counterclockwise by θ = 45 degrees (π/4 radians).

1. Apply the rotation transformation to the coordinates (x, y): $�=�\cos (�\mathrm{/}4)-�\sin (�\mathrm{/}4)=\frac{1}{\sqrt{2}}(�-�)$ $�=�\sin (�\mathrm{/}4)+�\cos (�\mathrm{/}4)=\frac{1}{\sqrt{2}}(�+�)$

2. Substitute these new coordinates into the ellipse equation: $4{(\frac{1}{\sqrt{2}}(�-�))}^2+9{(\frac{1}{\sqrt{2}}(�+�))}^2=36$

3. Simplify the equation to obtain the new representation of the ellipse: $\frac{1}{2}{�}^2+\frac{1}{4}{�}^2=1$

The rotated ellipse has a new representation in terms of (u, v), where the equation is simplified.

Example 2: Rotating a Hyperbola

Given the equation of a horizontally oriented hyperbola: $9{�}^2-16{�}^2=144$

To eliminate the cross terms and align the axes with the major and minor axes, we rotate counterclockwise by θ = 30 degrees (π/6 radians).

1. Apply the rotation transformation to the coordinates (x, y): $�=�\cos (�\mathrm{/}6)-�\sin (�\mathrm{/}6)=\frac{\sqrt{3}}{2}�-\frac{1}{2}�$ $�=�\sin (�\mathrm{/}6)+�\cos (�\mathrm{/}6)=\frac{1}{2}�+\frac{\sqrt{3}}{2}�$

2. Substitute these new coordinates into the hyperbola equation: $9{(\frac{\sqrt{3}}{2}�-\frac{1}{2}�)}^2-16{(\frac{1}{2}�+\frac{\sqrt{3}}{2}�)}^2=144$

3. Simplify the equation to obtain the new representation of the hyperbola: ${�}^2-{�}^2=4$

The rotated hyperbola has a new representation in terms of (u, v), where the equation is simplified.

These examples demonstrate how to find a new representation of a conic section equation after rotating it through a specified angle to simplify the equation and align the axes with the major and minor axes of the conic section.

To express equations of conic sections after rotation, you use the rotation of axes technique. This technique allows you to align the coordinate system with the major and minor axes of the conic section. The general equations for rotating the coordinates (x, y) through an angle θ counterclockwise are:

$�=�\cos (�)-�\sin (�)$ $�=�\sin (�)+�\cos (�)$

Once you have rotated the coordinates, you substitute these new coordinates (u, v) into the standard form equation of the conic section in the original coordinate system. After simplification, the equation will be in terms of (u, v).

Here are the equations for different types of conic sections in the rotated coordinate system:

1. Rotated Equation of an Ellipse: After rotating, the standard equation of an ellipse centered at the origin becomes:

$\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1$

Where (a, b) are the lengths of the major and minor axes, and (u, v) are the new coordinates.

2. Rotated Equation of a Hyperbola: After rotating, the standard equation of a hyperbola centered at the origin becomes:

$\frac{{�}^2}{{�}^2}-\frac{{�}^2}{{�}^2}=1$

Where (a, b) are the lengths of the transverse and conjugate axes, and (u, v) are the new coordinates.

3. Rotated Equation of a Parabola (Horizontal Axis): After rotating, the standard equation of a parabola with a horizontal axis and vertex at the origin becomes:

$4��={�}^2$

Where p is the distance between the vertex and the focus, and (u, v) are the new coordinates.

4. Rotated Equation of a Parabola (Vertical Axis): After rotating, the standard equation of a parabola with a vertical axis and vertex at the origin becomes:

$4��={�}^2$

Where p is the distance between the vertex and the focus, and (u, v) are the new coordinates.

These equations represent the conic sections in their rotated form, which simplifies the analysis and graphing of these curves when the axes of the conic sections are not aligned with the coordinate axes. The values of a, b, and p are specific to the particular conic section and may need to be determined based on the original standard form equation and the angle of rotation.

Writing equations of rotated conic sections in standard form involves expressing the conic sections in their original orientation with respect to the coordinate axes. You do this by applying the reverse of the rotation of axes transformation to bring the equation back to the standard form in terms of the original coordinates (x, y). Here are the standard forms for different types of conic sections and examples of how to write equations of rotated conics in standard form:

1. Standard Form of an Ellipse:

The standard form of an ellipse centered at the origin is:

$\frac{{�}^2}{{�}^2}+\frac{{�}^2}{{�}^2}=1$

where $�$ and $�$ are the lengths of the major and minor axes, respectively.

Example: Writing the Equation of a Rotated Ellipse in Standard Form

Suppose you have the rotated equation of an ellipse: $4{�}^2-6��+7{�}^2=1$

To bring it back to the standard form, reverse the rotation by finding the original coordinates $(�,�)$ in terms of the new coordinates $(�,�)$. The equations for the reverse rotation are:

$�=�\cos (�)+�\sin (�)$ $�=-�\sin (�)+�\cos (�)$

Here, $�$ is the angle of rotation. In this example, the original orientation is $�=30\mathrm{°}$ (π/6 radians).

Substitute these equations into the rotated ellipse equation: $4(�\cos (�\mathrm{/}6)+�\sin (�\mathrm{/}6){)}^2-6(�\cos (�\mathrm{/}6)+�\sin (�\mathrm{/}6))(�\sin (�\mathrm{/}6)-�\cos (�\mathrm{/}6))+7(�\sin (�\mathrm{/}6)-�\cos (�\mathrm{/}6){)}^2=1$

Simplify this equation to obtain the standard form equation.

2. Standard Form of a Hyperbola:

The standard form of a hyperbola centered at the origin is:

$\frac{{�}^2}{{�}^2}-\frac{{�}^2}{{�}^2}=1$

where $�$ and $�$ are the lengths of the transverse and conjugate axes, respectively.

Example: Writing the Equation of a Rotated Hyperbola in Standard Form

Suppose you have the rotated equation of a hyperbola: $3{�}^2-4��+2{�}^2=1$

To bring it back to the standard form, reverse the rotation using the original coordinates $(�,�)$ in terms of the new coordinates $(�,�)$, as in the previous example. The reverse rotation equations are the same.

Substitute these equations into the rotated hyperbola equation and simplify it to obtain the standard form equation.

3. Standard Form of a Parabola:

The standard form of a parabola with a vertical axis of symmetry and vertex at the origin is: $4��={�}^2$

The standard form of a parabola with a horizontal axis of symmetry and vertex at the origin is: $4��={�}^2$

Here, $�$ is the distance between the vertex and the focus.

Example: Writing the Equation of a Rotated Parabola in Standard Form

Suppose you have the rotated equation of a parabola with a vertical axis of symmetry: $3{�}^2-6��+2{�}^2=4�$

To bring it back to the standard form, reverse the rotation using the original coordinates $(�,�)$ in terms of the new coordinates $(�,�)$, as in the previous examples.

Substitute these equations into the rotated parabola equation and simplify it to obtain the standard form equation.

These examples demonstrate how to write the equations of rotated conic sections in standard form by reversing the rotation transformation. The key is to find the original coordinates $(�,�)$ in terms of the new coordinates $(�,�)$ using the reverse rotation equations.

You can identify conic sections without explicitly rotating axes by examining the coefficients and variables in the given equation. Here's how to recognize conics based on their general equation forms:

1. Ellipse: The general equation of an ellipse is:

$�{�}^2+�{�}^2+��+��+�=0$

To identify an ellipse:

• The coefficients of x^2 and y^2 must have the same sign.
• The coefficients of x^2 and y^2 must not be equal.
• The coefficients of x and y (C and D) must be zero.

Example:

• $4{�}^2+9{�}^2-36=0$ represents an ellipse.

2. Hyperbola: The general equation of a hyperbola is:

$�{�}^2-�{�}^2+��+��+�=0$

To identify a hyperbola:

• The coefficients of x^2 and y^2 must have opposite signs (A is positive, B is negative, or vice versa).
• The coefficients of x and y (C and D) must not be both zero.

Example:

• $9{�}^2-16{�}^2-144=0$ represents a hyperbola.

3. Parabola: The general equation of a parabola is:

$�{�}^2=��+�{�}^2+��+�$

To identify a parabola:

• Either the coefficient of x^2 (C) or the coefficient of y^2 (A) must be zero, but not both.

Example:

• $3{�}^2=4�+2{�}^2-5�+6$ represents a parabola.

By examining the coefficients and the variables in the equation, you can determine whether it represents an ellipse, hyperbola, or parabola without the need to rotate axes. If the equation does not match any of these patterns, it may represent a degenerate case, such as a point, line, or pair of intersecting lines.

Using the discriminant is a useful technique to identify and classify conic sections based on their equations. The discriminant helps you determine the type and orientation of a conic section. Here's how to use the discriminant to identify a conic:

1. General Conic Section Equation:

The general equation for a conic section in standard form is:

$�{�}^2+�{�}^2+��+��+�=0$

2. Calculate the Discriminant (D):

The discriminant (D) for the general conic section equation is calculated as follows:

$�=��-{\left(\frac{�}{2}\right)}^2$

3. Analyze the Discriminant:

Based on the value of the discriminant (D), you can identify the type and orientation of the conic section:

• If $�>0$:
  • If $�$ and $�$ have the same sign (both positive or both negative), it's an ellipse.
  • If $�$ and $�$ have opposite signs (one positive and one negative), it's a hyperbola.
• If $�=0$, it's a parabola.
• If $�<0$, the equation represents an imaginary or degenerate conic section.

Example 1: Identifying an Ellipse or Hyperbola

Consider the equation: $4{�}^2+9{�}^2-36=0$

Calculate the discriminant: $�=4\cdot 9-{\left(\frac{0}{2}\right)}^2=36$

Since $�>0$ and the coefficients of ${�}^2$ and ${�}^2$ have the same sign (both positive), the equation represents an ellipse.

Example 2: Identifying a Parabola

Consider the equation: $3{�}^2=4�+2{�}^2-5�+6$

Calculate the discriminant: $�=0\cdot 3-{\left(\frac{(-5)}{2}\right)}^2=\frac{25}{4}$

Since $�=0$, the equation represents a parabola.

Using the discriminant is a quick and effective way to identify and classify conic sections without needing to explicitly rotate axes. It provides information about the type and orientation of the conic based on the characteristics of the equation.