Problem 30 - 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. $$4 x^{2}-12 x y+9 y^{2}+\sqrt{6} x-29 y=91$$

Step-by-Step Solution

Answer

The given equation is of a rotated ellipse, and through calculations, it is determined that the axes are rotated by an angle of 67.38°. The graph of the given equation can be obtained by inputting the equation into a graphing utility, setting the appropriate window, and generating the graph.

Step 1: Identify the Type of Conic

The given equation is a form of conic section which is known as an ellipse by looking at the coefficients of the terms. An equation for a rotated ellipse has both \(x^{2}\) and \(y^{2}\) terms with the same sign but non-identical coefficients. Here, the coefficients of \(x^{2}\) is 4 and \(y^{2}\) is 9 which are both positive, but not identical.

Step 2: Calculate the Rotation of the Axis

The angle of rotation \(θ\) can be calculated by using the formula \(\frac{atan(2b/(a - c))}{2}\) where \(a\), \(b\) and \(c\) are the coefficients of \(x^{2}\), \(xy\) and \(y^{2}\) terms respectively. Substituting the values \(a = 4\), \(b = -6\), \(c = 9\) into the formula gives \(\frac{atan(2 * -6 /(4 - 9))}{2} = atan(-12 / -5) = atan(2.4) = 67.38°\). So, the axes are rotated through 67.38°.

Step 3: Using a Graphing Utility to Plot the Graph

Firstly, input the equation \(4 x^{2}-12 x y+9 y^{2}+\sqrt{6} x-29 y=91\) into the graphing utility. Then, set the window to include all important parts of the graph and possible rotations due to the angle calculated in the previous step. After that, press the enter or the graph button depending on the utility to generate the graph. It is advisable to use zoom in or zoom out options to ensure all details are captured and to adjust the viewing window as necessary.

Step 4: Interpreting the Graph

The graph will represent an ellipse because of the coefficients of the equation. The rotation of the axes can be observed visually on the graph by noting the direction of major and minor axes of the ellipse.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a right circular cone with a plane. They include circles, ellipses, parabolas, and hyperbolas. Each type of conic section has a unique set of algebraic properties that allow us to distinguish one from another.

For instance, the general equation of a conic section is given by the second-degree polynomial equation \(Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0\). When graphing conic sections, one can identify an ellipse when both squared terms, \(x^{2}\) and \(y^{2}\), have the same sign and their coefficients are not equal - precisely like the equation given in our exercise. By understanding these sign and coefficient relationships, you can swiftly discern the nature of the conic even before plotting.

Graphing Utility Use

Utilizing a graphing utility is a crucial skill for students navigating through precalculus and beyond. These digital tools allow for swift visualization of complex equations, aiding in understanding and analysis. To graph an equation like \(4 x^{2}-12 x y+9 y^{2}+\sqrt{6} x-29 y=91\), one typically inputs the equation into the software, adjusting the graphing window to ensure a proper view of the curve.

Graphing utilities often feature functions like zoom, trace, and window adjustment, enabling detailed examination of the graph's shape, intercepts, and asymptotes. Some advanced software can even animate the rotation of conic sections for a deeper grasp of the concept. When studying rotated ellipses, using these utilities makes it more manageable to comprehend the transformation and orientation of axes.

Axis Rotation Calculation

The axis rotation calculation is an essential concept when dealing with rotated conic sections, as it determines the angle through which the original axes have been tilted. For a generalized conic equation \(Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0\), where \(B \eq 0\), suggests a rotation. To find the rotation angle \(\theta\), we use the formula \(\theta = \frac{\arctan(2B/(A - C))}{2}\).

For our rotated ellipse equation, this calculation was applied to find the angle of 67.38°. It's crucial to understand this step as it aids in visualizing the rotated ellipse on a graph and can influence how one might approach problems involving properties such as eccentricity, foci, and directrices in the context of the rotated axes.

Ellipses in Precalculus

Ellipses are a fundamental part of the study of conic sections in precalculus. Understanding their equations, which are generally given by \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) for a horizontal major axis, or \(\frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}} = 1\) for a vertical major axis, is crucial. When these ellipses are rotated, they take a more complex form, as seen in the given exercise.

Grasping the features of ellipses, like their major and minor axes, foci, and center, supports conceptual comprehension and prepares students for later topics such as optimization, orbital mechanics, and mathematical modeling. Through practice and visualization with the aid of graphing utilities, students can demystify the nature of rotated ellipses and excel in their studies.