Problem 27 - 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. \(x^2+2xy+y^2=20\)

Step-by-Step Solution

Answer

First, identify the coefficients in the general rotated conic equation to determine that the graph represents a rotated ellipse. Then graph the equation with a graphing utility and use the equation \(\theta = \frac{1}{2} \text{atan}(B/(A-C))\) to calculate the angle of rotation \(\theta\).

Step 1: Understanding the equation

The equation given is \(x^2+2xy+y^2=20\). This is the equation of a rotated conic. The presence of the \(xy\) term indicates that this conic section (a circle, parabola, ellipse or hyperbola) is not aligned with the x and y axes but is rotated by some angle \(\theta\).

Step 2: Graphing the equation

Use a graphing utility to plot the equation \(x^2+2xy+y^2=20\). The graph you get should be a rotated ellipse. Rotation does not change the shape of the conic, only its orientation.

Step 3: Finding the angle of rotation

The angle of rotation \(\theta\) for a rotated conic with equation \(Ax^2 + Bxy + Cy^2 = 1\) (in this case, \(A = C = 1, B = 2\), and the conic is scaled up to = 20), can be found using the following equation: \(\theta = \frac{1}{2} \text{atan}(B/(A-C))\). Thus, substituting \(A = C = 1\) and \(B = 2\) into the angle of rotation formula gives \(\theta = \frac{1}{2} \text{atan}(2)\). Use a calculator to evaluate this to get \(\theta\).

Step 4: Interpreting the result

The calculated value of \(\theta\) is the counterclockwise angle from the positive x-axis to the major axis of the rotated ellipse. It represents the amount the original axes used to describe the ellipse have been rotated to produce the given equation.

Key Concepts

Angle of Rotation

To understand the concept of angle of rotation, consider that some conic sections are inclined or rotated rather than aligned along the x and y axes. This rotation is indicated by the presence of an \(xy\) term in the equation. For the conic section given, the equation is \(x^2+2xy+y^2=20\). The formula to find the angle of rotation \(\theta\) involves the coefficients of the terms in the general conic equation. Specifically, the formula is \[ \theta = \frac{1}{2} \tan^{-1}\left(\frac{B}{A-C}\right) \] In this expression, \(A\), \(B\), and \(C\) correspond to the coefficients in the equation \(Ax^2 + Bxy + Cy^2 = D\). By substituting \(A = 1\), \(B = 2\), and \(C = 1\), we find the angle as \[ \theta = \frac{1}{2} \tan^{-1}(2) \] Thus, the angle of rotation aligns the major axis of the ellipse in its rotated position.

Graphing Utility

Utilizing a graphing utility can greatly simplify visualizing complex equations such as the conic sections. A graphing utility helps by:

• Providing a visual representation which aids in identifying the type of conic section.
• Allowing for direct input of equations with terms like \(xy\), which indicate rotations or tilts.
• Offering tools to zoom, translate, or rotate the view as necessary for better analysis.

For the equation \(x^2+2xy+y^2=20\), entering it into a graphing calculator or software can directly show the resulting shape, in this case, a rotated ellipse. Pay attention to options for axis rotation display, which can help visualize the effect of the angle of rotation on the graph.

Ellipse

An ellipse is a type of conic section that appears as an elongated circle. It is defined by the sum of the distances from any point on the ellipse to two fixed points (the foci) being constant. In the context of the given problem, the equation \(x^2+2xy+y^2=20\) describes an ellipse because it features terms that indicate it's not positioned along the principal axes.
The rotation does not alter the fundamental nature of the ellipse, but it changes its orientation. A graphing utility will show this as a stretched circle (ellipse) that doesn't align perfectly with the horizontal and vertical axes, due to its rotation denoted by the \(xy\) term.

XY-term

The existence of an \(xy\) term in a conic section equation like \(x^2+2xy+y^2=20\) is a distinctive marker that tells us more than just the type of conic. It signifies that the conic is rotated. This rotation angle \(\theta\) can be mathematically deduced, as previously explained, using the formula for the angle of rotation.
Without this \(xy\) term, the conic would have its axes aligned with the x and y axes. Therefore, the presence of the \(xy\) term requires additional calculations and visualization to understand how the conic is situated in its own coordinate system relative to the primary axes.