Problem 36 - Precalculus with Limits

Question

In Exercises 27-36, use a graphing utility to graph the conic. Determine the angle \(\theta\) through which the axes are rotated.Explain how you used the graphing utility to obtain the graph. \(6x^2-4xy+8y^2+(5\sqrt{5} - 10)x - (7\sqrt{5}+5)y=80\)

Step-by-Step Solution

Answer

The angle of rotation of the axes of the conic is approximately \(63.43^{\circ}\)

Step 1: Graph the conic

Use a graphing utility to graph the given conic equation \(6x^2-4xy+8y^2+(5\sqrt{5} - 10)x - (7\sqrt{5}+5)y=80\). The shape, orientation, and location of the conic section on the graph will be visible from this step.

Step 2: Identify the coefficients

Identify the coefficients A, B, and C in the equation. Here, A = 6, B = -4, and C = 8.

Step 3: Calculate the angle of rotation

Use the rotation angle formula \(\theta = \frac{1}{2} atan\left(\frac{B}{A-C}\right)\) to calculate the angle of rotation. Substituting A = 6, B = -4, and C = 8 in the formula to get \(\theta = \frac{1}{2} atan\left(\frac{-4}{6-8}\right) = atan(2) = 63.43^{\circ}\).

Key Concepts

Angle of Rotation

The angle of rotation is essential when graphing conic sections, especially if the axes are tilted. When a conic section is rotated, its appearance on the graph changes, making it necessary to calculate this angle to understand its orientation better. For our given conic equation, the angle of rotation helps transform the equation into one that's easier to analyze.

The angle of rotation, denoted as \( \theta \), can be determined from the coefficients of the conic equation using the formula:

• \( \theta = \frac{1}{2} \tan^{-1}\left(\frac{B}{A-C}\right) \)

Here, \( A \), \( B \), and \( C \) are the coefficients from the general quadratic form of the conic. In our exercise, substituting \( A = 6 \), \( B = -4 \), and \( C = 8 \) into the formula gives us \( \theta = \tan^{-1}(2) = 63.43^{\circ} \).

Understanding the angle of rotation allows you to transform the coordinate axes, thus simplifying the analysis of the conic section. This rotational adjustment aligns the graph with the principal axes, offering a more straightforward interpretation of its properties.

Conic Sections

Conic sections form a fundamental concept in geometry, representing the curves obtained by intersecting a cone with a plane. These include circles, ellipses, parabolas, and hyperbolas. Each type of conic section has distinct characteristics determined by the equation's coefficients.

In the given exercise, the presence of the \( xy \) term and the coefficients suggests a rotated conic section. The equation is written in its general form:

• \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \)

The \( Bxy \) term indicates that rotation of the conic section has likely occurred. It implies that the axes need to be reoriented to achieve the standard form. By examining the values of \( A \), \( B \), and \( C \), and applying the rotation formulas, the conic can be classified and its properties understood.

Mastering conic sections involves recognizing these forms and transformations, essential for solving equations and predicting real-world scenarios. Whether in physics, engineering, or architecture, conic sections are present, and proficiency in manipulating these equations is crucial.

Graphing Utility Usage

Graphing utilities are powerful tools that aid in visualizing complex conic sections. They allow you to plot equations accurately, observe transformations, and identify key features not easily seen algebraically. Using a graphing utility simplifies the process of graphing, rotating, and analyzing conic sections.

To graph the given conic equation, enter the expression \( 6x^2-4xy+8y^2+(5\sqrt{5} - 10)x - (7\sqrt{5}+5)y = 80 \) into the utility. The tool will display the conic's graph, showing its shape and location. Advanced utilities might also allow you to perform transformations such as rotations, making it easier to visualize how the conic is oriented compared to the axes.

• Step 1: Input the equation into the graphing utility.
• Step 2: Adjust settings for scale and range, ensuring the entire conic is visible.
• Step 3: Analyze the graph for symmetry, direction, and asymptotes if applicable.

By utilizing a graphing utility, you gain insights that facilitate understanding complex relationships within conic sections, enhancing both learning and practical application.