Problem 46 - Calculus Early Transcendentals: Pearson New International Edition

Question

Eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form. Finally, graph the equation showing the rotated axes. 4 x y-3 y^{2}=64

Step-by-Step Solution

Answer

Rotate the axes to eliminate \( xy \) term and graph accordingly.

Step 1: Identifying the Coefficients

First, identify the coefficients of the given quadratic equation. The original equation is \( 4xy - 3y^2 = 64 \). Here, the coefficients are: \( A = 0 \), \( B = 4 \), \( C = -3 \).

Step 2: Finding the Rotation Angle

Compute the angle \( \theta \) for the rotation to eliminate the cross-product term. Use the formula \( \tan 2\theta = \frac{B}{A - C} \). Substituting the values, \( \tan 2\theta = \frac{4}{0 + 3} = \frac{4}{3} \). Solve for \( \theta \).

Step 3: Computing Cosine and Sine of 2θ

Once \( \tan 2\theta \) is known, find \( \cos 2\theta \) and \( \sin 2\theta \) using trigonometric identities. If \( \tan 2\theta = \frac{4}{3} \), then \( \sin 2\theta = \frac{4}{5} \) and \( \cos 2\theta = \frac{3}{5} \).

Step 4: Apply the Rotation of Axes

To eliminate the \( xy \) term, substitute the rotated variables: \( x = x'\cos \theta - y'\sin \theta \) and \( y = x'\sin \theta + y'\cos \theta \). Substitute \( \cos \theta = \sqrt{\frac{5 + 3}{10}} \) and \( \sin \theta = \sqrt{\frac{5 - 3}{10}} \). Then simplify the equation in terms of \( x' \) and \( y' \).

Step 5: Rewrite the Equation in Standard Form

The new equation should be rewritten without the \( x'y' \) term. To complete the square if necessary, ensure that the terms in \( x'^2 \) and \( y'^2 \) match the form of a conic section, such as an ellipse, hyperbola, or parabola.

Step 6: Plot the Equation

Graph the final equation in new coordinates \( x' \) and \( y' \). Ensure the graph represents the correct orientation after rotation, showing axes.

Key Concepts

Cross-Product Term

In a quadratic equation involving two variables, a cross-product term is the term that involves the multiplication of these variables, typically represented as \( Bxy \). This term complicates the classification and graphing of conic sections when dealing with equations in their general form.
To simplify the equation, we eliminate this term using a mathematical process known as rotation of axes.
By rotating the coordinate axes, we align them in such a way that the cross-product term disappears from the equation, making it easier to identify and graph the conic section involved.

• To eliminate the cross-product term, calculate the rotation angle \( \theta \) using the formula: \( \tan 2\theta = \frac{B}{A - C} \).
• After rotation, the new axes will be denoted as \( x' \) and \( y' \).

This transformation simplifies the process of determining the type of conic section represented by the equation.

Quadratic Equation

A quadratic equation in two variables usually takes the form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). The presence of the \( xy \) term can make analyzing and graphing the equation more challenging.
Eliminating this term via rotation transforms the equation into a simpler form, where the analysis can focus on the coefficients of \( x'^2 \), \( y'^2 \), and the constant term.

• Quadratic equations can represent different conic sections, such as ellipses, parabolas, and hyperbolas, depending on the coefficient values.
• By reducing the equation into standard form without the cross-product term, it becomes easier to identify the specific type of conic represented and graph the equation correctly.

Conic Sections

Conic sections are the shapes created when a double cone is sliced through with a plane. They are essential in various mathematical and scientific fields. The primary conic sections are:

• Ellipse: Appears when the plane cuts across the cone at an angle, creating an oval shape.
• Parabola: Formed when the plane is parallel to the slope of the cone, resulting in a U-shaped curve.
• Hyperbola: Occurs when the plane cuts both cones, producing two separate curves.

By rotating the axes and removing the cross-product term, a quadratic equation can be rewritten to clearly display one of these familiar conic sections. Identifying the type of conic section helps predict the graph's shape and orientation, facilitating easier interpretation and graphing of the equation. This process supports an accurate depiction of geometric relationships in a wide range of applications.

Trigonometric Identities

Trigonometric identities are mathematical equations that express certain relationships between the trigonometric functions. They are instrumental in simplifying the rotation of axes process.

• In the context of rotating axes to eliminate the \( xy \) term, identities such as \( \sin^2 \theta + \cos^2 \theta = 1 \) provide the necessary simplification to determine \( \cos 2\theta \) and \( \sin 2\theta \).
• Once \( \tan 2\theta \) is calculated, these identities help derive \( \cos \theta \) and \( \sin \theta \), which allow us to transform the original equation by substituting \( x = x'\cos \theta - y'\sin \theta \) and \( y = x'\sin \theta + y'\cos \theta \).

These transformations result in a rotated coordinate system that aligns with one of the principal axes of the conic section, facilitating an easier analysis and graphing of the quadratic equation.