Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, determine the angle \(\theta\) that will eliminate the \(x y\) term and write the corresponding equation without the \(x y\) term. \(16 x^{2}+24 x y+9 y^{2}+6 x-6 y+2=0\)

Step-by-Step Solution

Answer

Transform the equation using \(\theta\) where \(\tan 2\theta = \frac{24}{7}\).

Step 1: Identify the Coefficient of the xy term

For the given equation, the coefficient of the \(xy\) term is 24. We denote this coefficient as \(B = 24\).

Step 2: Use the Formula for Eliminating xy-Term

To eliminate the \(xy\) term, use the rotation transformation formula. The angle \(\theta\) is found using the formula \(\tan 2\theta = \frac{B}{A - C}\), where \(A = 16\), \(B = 24\), and \(C = 9\).

Step 3: Calculate tan(2θ)

Substitute the values into the equation \(\tan 2\theta = \frac{24}{16 - 9}\). This results in \(\tan 2\theta = \frac{24}{7}\).

Step 4: Determine θ

To find \(\theta\), calculate \(2\theta\) using \(\tan^{-1}(\frac{24}{7})\). Then, divide the result by 2 to find \(\theta\).

Step 5: Write the Equation without xy-term

Using the calculated \(\theta\), transform the original equation into one where the \(xy\) term is eliminated using rotation matrices or symbolic software.

Key Concepts

Rotation of Axes

In coordinate geometry, the rotation of axes is a crucial technique that allows us to simplify conic sections by eliminating the mix term, known as the \( xy \) term. Sometimes, conic sections appear in complicated forms because of this term, making it challenging to identify the type of conic, like an ellipse, parabola, or hyperbola. By rotating the axes, we aim to cancel out the \( xy \) term, turning our equation into one of the standard forms.When we decide to rotate the axes, we introduce a new angle, \( \theta \), which lets us realign our coordinate system such that the conic section can be expressed without the \( xy \) cross-product term. The new equations become much easier to work with and evaluate.

• The concept involves replacing \( x \) and \( y \) with new coordinates \( x' \) and \( y' \) based on trigonometric transformation formulas.
• The formulas used include \( x' = x\cos \theta + y\sin \theta \) and \( y' = -x\sin \theta + y\cos \theta \).

Let's put it this way: think of your graph spinning around the origin point until the mixed \( xy \) term fades away. This rotation ensures the conic section aligns perfectly with the axes, allowing clearer insights into its properties.

Quadratic Forms

Quadratic forms, especially in the context of conic sections, involve equations that are quadratic with respect to variables \( x \) and \( y \). These forms are usually written as \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). This type of equation can describe a variety of geometric shapes when plotted on a coordinate plane.The quadratic form helps determine the nature of the curve, whether it is an ellipse, a parabola, or a hyperbola. This nature often depends significantly on the values of coefficients \( A \), \( B \), and \( C \), particularly the \( B \) value which introduces the \( xy \) term, indicating the rotation effect on the conic shape.

• The term \( Ax^2 + Cy^2 \) defines the basic quadratic nature of the curve.
• The appearance of \( Bxy \) means the axes are not aligned with the conic section's principal axes.
• Rotation of axes effectively reduces this equation to one without the \( xy \) term, simplifying further analysis or problem-solving steps.

Think of the \( Bxy \) term as a clue that tells us what adjustments (like rotating the axes) are necessary to view the conic section in its simplest or most recognizable form.

Angle of Rotation

Determining the angle of rotation is essential when we aim to simplify quadratic equations with mixed terms. The angle helps achieve a clearer representation by eliminating the \( xy \) component, making it easier to identify the specific conic.The formula to find the required angle \( \theta \) uses the trigonometric relationship \( \tan 2\theta = \frac{B}{A - C} \), where \( A \), \( B \), and \( C \) are coefficients from the quadratic form. By calculating \( 2\theta \) and subsequently \( \theta \), one can apply this angle to transform the coordinates.Here’s a simple breakdown of how it works:

• Calculate \( \tan 2\theta \) using the coefficients from the original equation.
  
• Use the inverse tangent function to determine \( 2\theta \).
  
• Finally, divide by 2 to isolate \( \theta \), which represents the rotation needed.

Consider this angle as the key to "unlocking" the simplest viewing angle for the conic structure. The process acts like fine-tuning a camera lens, allowing us to view mathematical structures from the clearest possible perspective.