Problem 33 - 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. \(2x^2+4xy+2y^2+\sqrt{26}x+3y=-15\)

Step-by-Step Solution

Answer

The angle through which the axes are rotated is 45 degrees.

Step 1: Identify coefficients

Firstly, identify the coefficients for \(a\), \(b\), and \(c\) from the given equation. Here, \(a=c=2\), and \(b=4\).

Step 2: Compute rotation angle

Once the coefficients are identified, use the formula \(\cot {2\theta} = \frac{a-c}{b}\) to find the rotation angle. Since \(a=c\), \(\cot {2\theta} = 0\), which occurs when \(\theta = \frac{\pi}{4}\) or \(\frac{3\pi}{4}\). Because of the positive sign of \(b\), we take \(\theta = \frac{\pi}{4}\) (45 degrees).

Step 3: Graph the conic

Graph the conic using a graphing utility by plugging in the given equation. Observe the rotation of the conic.

Step 4: Use graphing utility

Now, rotate the axes in the graphing utility by -45 degrees and you'll see the new graph of the conic. Here, the axes are along the principal directions and there would be no cross term \(xy\) in the new equation.

Key Concepts

Rotation of Axes

Conic sections, like ellipses, parabolas, and hyperbolas, often appear rotated in the plane. Understanding the rotation of axes is essential to simplifying these equations and revealing their true nature. To eliminate the cross term like the one present (xy), we rotate the coordinate axes by a specific angle. This transformation simplifies the equation by aligning it along the principal axes.

The mathematical process involves identifying the coefficients of the quadratic terms, and calculating the angle \(\theta\) through which the axes need to be rotated. Using the formula \(\cot{2\theta} = \frac{a-c}{b}\), where \(a\), \(b\), and \(c\) come from the terms of the equation, we find \(\theta\). In scenarios where \(a=c\), as in your exercise, the formula simplifies since \(\cot{2\theta} = 0\). This condition means a 45-degree rotation, which eliminates the \(xy\) term, simplifying the equation and helping to analyze the graph more effectively.

Graphing Utility

Graphing utilities are powerful tools for visually representing equations, including rotated conic sections. They provide an interactive way to explore the characteristics of these curves, making complex mathematical concepts more accessible.
Here's how you can utilize a graphing utility to graph conic sections like the one in your exercise:

• Input the original equation into the graphing utility. This helps in visualizing the conic's current state and understanding its orientation due to the \(xy\) term.
• The graph will likely appear tilted due to the presence of this cross term.
• Once the graph is plotted, most graphing utilities allow for manipulation of the axes. You can rotate the axes by the calculated angle, which, in this case, is -45 degrees.
• Observing the newly altered graph post-rotation, you will notice a cleaner representation aligned along the principal axes with no cross term present.

This method doesn't only help with visual understanding but also confirms the accuracy of analytical solutions based on rotations. It bridges algebraic thought with visual geometry.

Angle of Rotation

The angle of rotation is a crucial element in transforming the equation of a conic section to eliminate the \(xy\) term, leading to a simpler expression.

In the given exercise, finding out the exact angle involves a bit of careful calculation. By using the formula \(\cot{2\theta} = \frac{a-c}{b}\), you determine the angle \(\theta\) which allows the conic to appear directly aligned with the axes. This was found to be \(45\) degrees in your case.

• Recognize that this angle effectively neutralizes the skew introduced by the \(xy\) term.
• The transformation simplifies the equation and makes it easier to understand the inherent characteristics of the conic.
• Understanding this angle isn't just about converting one form to another, but also about unlocking clear geometric insights that are important for deeper comprehension.

Comprehension of angles of rotation offers students a pivotal skill, balancing both calculation and perception, regarding how such transformations simplify complex mathematical landscapes.