Problem 21 - College Algebra

Question

For the following exercises, find a new representation of the given equation after rotating through the given angle. $$-2 x^{2}+8 x y+1=0, \theta=45^{\circ}$$

Step-by-Step Solution

Answer

The new representation is \(3x'^2 - 3y'^2 + 1 = 0\).

Step 1: Identify the Rotation Matrix

To rotate the coordinate system by an angle \( \theta = 45^{\circ} \), we use the rotation matrix: \[ R = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} \].

Step 2: Apply the Rotation to the Variables

Substitute the old coordinates \((x, y)\) with the new ones \((x', y')\) using the equations derived from the rotation matrix: \[ x = \frac{\sqrt{2}}{2}(x' - y') \] and \[ y = \frac{\sqrt{2}}{2}(x' + y') \].

Step 3: Substitute and Simplify the Equation

Substitute \( x \) and \( y \) in the original equation \(-2x^2 + 8xy + 1 = 0\) with the expressions for \( x \) and \( y \) in terms of \( x' \) and \( y' \). Simplify this new equation: \[ -2 \left(\frac{\sqrt{2}}{2}(x' - y')\right)^2 + 8 \left(\frac{\sqrt{2}}{2}(x' - y')\right) \left(\frac{\sqrt{2}}{2}(x' + y')\right) + 1 = 0 \]Perform the expansion and simplification to get the equation in terms of \( x' \) and \( y' \).

Step 4: Complete the Simplification

Finish the algebraic simplification process, combining terms and simplifying using algebraic identities (such as multiplication, combining like terms, etc.). You should arrive at an equation of the form: \[ 3x'^2 - 3y'^2 + 1 = 0 \].

Step 5: Interpret the Results

This new equation represents the original conic section rotated by \( 45^{\circ} \). The terms \( x'^2 \) and \( y'^2 \) reflect the changes caused by the rotation, with the coefficients showing the new relationship between these variables.

Key Concepts

Understanding the Rotation Matrix

The rotation matrix is a powerful tool in mathematics used to rotate the coordinate system. Imagine you have a picture and you want to turn it at a certain angle. Similarly, the rotation matrix helps in changing the direction of the axes in a coordinate system.
The rotation matrix for an angle \( \theta \) is given by:

• \( R = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \)

When \( \theta = 45^{\circ} \), the sines and cosines equate to \( \frac{\sqrt{2}}{2} \). This means the rotation matrix becomes:

• \( \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} \)

Using this matrix, you rotate the axes, transforming how each point in the plane is represented. Each point \((x, y)\) becomes \((x', y')\), a crucial transformation when simplifying equations like conics.

The Role of Coordinate Transformation

A coordinate transformation involves changing the coordinate system to simplify equations or visualize data from a different perspective. By rotating the axes using the rotation matrix, we find new coordinates \((x', y')\) from the old \((x, y)\).

• The relationship is given by:
• \( x = \frac{\sqrt{2}}{2}(x' - y') \)
• \( y = \frac{\sqrt{2}}{2}(x' + y') \)

This transformation adjusts the original position and orientation of points. Why do it? Because some mathematical problems, especially involving conic sections, become simpler when viewed from a different angle. Using these new coordinates, we rewrite the original equation into an easier form to work with.

Understanding Conic Sections in the Context of Rotation

Conic sections like ellipses, parabolas, and hyperbolas are curves obtained by slicing a cone. In algebraic terms, they are portrayed with equations like the given \(-2x^2 + 8xy + 1 = 0\).
Rotating these curves using transformations can alter their appearance and make them simpler to analyze.When we rotate a conic section using a 45° rotation, the coefficients of \( x \) and \( y \) change, altering their orientation. This is crucial in identifying their types or classes, as seen in the exercise where the result of the equation becomes a more straightforward quadratic in terms of \( x' \) and \( y' \).

Rotations help clear up complex relationships between variables initially tangled in each other like \( xy \) and linear terms.

Simplifying Through Algebraic Simplification

Algebraic simplification is the art of making expressions more manageable. It involves clearing up an equation by combining like terms, knocking out unnecessary complexity, and applying known identities.
After substituting the new variables, \( x' \) and \( y' \), the goal is to clear out extra terms through simplification.

• Expand each polynomial term, clear up fractions, and identify common factors.
• Use distributive properties correctly.

In this exercise, once the coordinates are transformed and substituted, the equation simplifies to \( 3x'^2 - 3y'^2 + 1 = 0 \).
This form is far less complex than the original, making it easier to understand and analyze the conic's properties. Algebraic simplification is a crucial phase that sharpens our mathematical understanding of transformed coordinates.