Problem 27 - Precalculus

Question

Use a graphing utility to graph the conic. Determine the angle 8 through which the axes are rotated. Explain how you used the graphing utility to obtain the graph. $$40 x^{2}+36 x y+25 y^{2}=52$$

Step-by-Step Solution

Answer

The angle for which the axes are rotated is calculated from the formula to be approximately 0.674741 degrees.

Step 1: Identify the type of Conic

The equation \(40x^{2} + 36xy + 25y^{2} = 52\) is a general form of a second degree equation in x and y. It represents a conic section, and looking at its form, it's clear that it is a rotated conic since the coefficient of xy term is non-zero.

Step 2: Plot the graph using graphing utility

To plot this equation, enter the equation \(40x^{2} + 36xy + 25y^{2} = 52\) in the graphing utility. Adjust the scale and window size as needed to get a clear view of the graph. This will give a clear view of how the conic section looks like and also visualize the rotation in the graph.

Step 3: Determine the rotation angle

From the graph, observe the tilt in the plotted curve. This tilt provides the angle of rotation. The rotation angle \(\theta\) can be found using the formula \(\theta = 0.5 * atan(2 * b / (a - c))\) where \(a = 40\), \(b = 36/2 = 18\) and \(c = 25\). Substituting these values gives \(\theta = 0.5 * atan(2 * 18 / (40 - 25))\), which simplifies to \(\theta = 0.5 * atan(36 / 15)\). Evaluate this expression to get the angle of rotation in degrees.

Key Concepts

Graphing Utilities Usage

Graphing utilities are invaluable tools for visualizing mathematical concepts, especially when dealing with complex figures like conic sections. These utilities allow for a rapid transition from an algebraic equation like \(40x^{2} + 36xy + 25y^{2} = 52\) to its graphical counterpart, which is crucial for spatial understanding. To employ a graphing utility effectively, input the given equation and carefully adjust the viewing window to capture the complete graph of the conic. Ensure that the scale is proportionate so that the shape is not distorted, which could mislead you when identifying the conic type and its rotation.

By zooming in and out and shifting the axes, a more precise picture emerges. This visual representation helps not only in identifying the conic section but can also hint at any rotational transformation it has undergone. As part of the exercise improvement advice, it's really beneficial to explore the features of your graphing utility, perhaps you can use sliders to dynamically understand the impact various coefficients have on the shape and orientation of the conic.

Conic Section Identification

Identifying a conic section from a general second-degree equation is essential to understanding its properties and behavior. A second-degree equation of the form \(Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0\) encompasses all conic sections; however, the presence and value of the \(Bxy\) term is the key indicator of a rotation. If \(B\) is non-zero, as in our case with \(36xy\), the conic section is rotated.

Without rotation, you can classify conics simply by comparing the coefficients of \(x^{2}\) and \(y^{2}\). For instance, if both are positive and equal with \(B = 0\), the conic is a circle; if one is negative, it's a hyperbola; unequal but same sign yields an ellipse; and if one coefficient is zero, it results in a parabola. Including these tips can assist students in identifying conic sections before proceeding to graph them.

Conic Section Rotation Angle

Once a rotated conic section is identified, determining the angle of rotation is crucial for analyzing its properties. The angle \(\theta\) represents the counterclockwise rotation necessary to eliminate the \(xy\) term and transform the equation into a standard form. The formula \(\theta = 0.5 * \text{atan}(2 * b / (a - c))\) is derived from trigonometric principles, where \(a\), \(b\), and \(c\) are the coefficients of the terms \(x^{2}\), \(xy\) (with \(b\) halved), and \(y^{2}\), respectively in the given equation.

Understanding and applying this formula allows students to discover the exact angle that the axis has been tilted. This insight reveals the original orientation of the conic section had it not been rotated, and it is pivotal in analyzing its properties related to the coordinate plane. It is advisable to practice with various equations to solidify the concept of rotation angle and its calculation.

Second-Degree Equation Analysis

Delving into the analysis of a second-degree equation, one must first acknowledge that these equations represent conic sections, the curves formed by the intersection of a plane and a double-napped cone. The general equation \(Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0\) requires careful inspection to determine both the nature of the conic and any rotational factors present.

Analysis begins with the coefficients of the squared terms and the cross-product term \(Bxy\). Zero value for \(B\) indicates no rotation, whereas a non-zero value calls for further investigation using algebraic methods or graphing utilities to visualize the conic's orientation. The discriminant \(B^{2} - 4AC\) is another useful element; its sign (positive, negative, or zero) sets the conic class apart—depending on whether you're dealing with a hyperbola, ellipse, or parabola. Mastery of analyzing these equations will provide students with a deeper understanding of conic sections and their various appearances in rotated forms.