Problem 16 - Precalculus with Calculus Previews

Question

In Problems \(11-16,\) use rotation of axes to eliminate the \(x y\) -term in the given equation. Identify the conic and graph. $$ x^{2}+4 x y+4 y^{2}=16 \sqrt{5} x-8 \sqrt{5} y $$

Step-by-Step Solution

Answer

Rotate axes to find the rewritten equation, classify it as an ellipse, and graph accordingly.

Step 1: Rewrite Equation in Standard Form

Start by rearranging the given equation to bring all terms to one side: \[ x^2 + 4xy + 4y^2 - 16\sqrt{5}x + 8\sqrt{5}y = 0 \] Recognize that this is a conic section.

Step 2: Determine Rotation Angle

Use the formula \( \tan 2\theta = \frac{B}{A-C} \) to find the rotation angle \( \theta \). Here, \( A = 1 \), \( B = 4 \), and \( C = 4 \).\[ \tan 2\theta = \frac{4}{1-4} = -\frac{4}{3} \] Calculate two possible angles where \( \tan 2\theta = -\frac{4}{3} \).

Step 3: Determine Rotation Angle (Continued)

Calculate \( \theta \) such that this new line of rotation eliminates the \( xy \)-term.Using \( \tan 2\theta = -\frac{4}{3} \), find \( \theta = \frac{1}{2} \tan^{-1}(-\frac{4}{3}) \).

Step 4: Apply Rotation Formulas

After finding \( \theta \), rewrite \( x \) and \( y \) in terms of new variables \( x' \) and \( y' \) using\[ x = x' \cos \theta - y' \sin \theta \]\[ y = x' \sin \theta + y' \cos \theta \].Substitute these into the given equation to eliminate the \( xy \)-term.

Step 5: Simplify and Classify the Conic Section

Simplify the resulting equation to identify the type of conic. The resulting equation should no longer contain the \( xy \)-term.

Step 6: Graph the Conic Section

Using the simplified form, identify the conic and plot it. Based on the equation, determine whether it's an ellipse, hyperbola, or parabola. Since the original equation was symmetrical and both \( A \) and \( C \) were positive, the conic is likely an ellipse.Graph it using the standard form.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. These include circles, ellipses, parabolas, and hyperbolas. Each type has a unique set of properties and equations. In general, conic sections can be represented by a second-degree equation in two variables:

• Standard form: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
• Unique identifiers:
  • If \( B^2 - 4AC < 0 \), it's an ellipse.
  • If \( B^2 - 4AC = 0 \), it's a parabola.
  • If \( B^2 - 4AC > 0 \), it's a hyperbola.

The equation given in the original exercise is a conic section that requires further manipulation to identify its specific type because it includes an \( xy \)-term.

Eliminating xy-term

An essential step in dealing with the general conic equation is to eliminate the \( xy \)-term using a rotation of axes. This is important because it simplifies classification and graphing. When a conic equation in the form of \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]has a non-zero \( B \)-term, you can rotate the axes to eliminate it. Here's how it's done:

• Determine the rotation angle \( \theta \) using \( \tan 2\theta = \frac{B}{A-C} \).
• Calculate \( \theta \) as \( \theta = \frac{1}{2} \tan^{-1}\left( \frac{B}{A-C} \right) \).
• Substitute back into the equations using:
  • \( x = x' \cos\theta - y' \sin\theta \)
  • \( y = x' \sin\theta + y' \cos\theta \)

Rotating the axes helps transform the equation into one with no \( xy \)-term, making it easier to identify and graph the conic.

Graphing Conics

Once the \( xy \)-term has been eliminated, graphing the conic section becomes straightforward. The new equation typically reveals the conic's nature more clearly, whether it’s an ellipse, parabola, or hyperbola. Follow these steps to plot the conic:

• Identify the conic based on the coefficients and the signs after simplification. An ellipse has positive coefficients for \( A \) and \( C \), a hyperbola has one positive and one negative, and a parabola has either \( A \) or \( C \) zero.
• Add details like the center, foci, axes, or directrix as appropriate for the conic type. It's also helpful to determine whether the axes need to be rescaled or shifted.
• Plot using the simplified equation, ensuring symmetry and key features are accurate. Use graph paper or graphing software for precision.

The graph provides a visual understanding of the conic's properties such as shape, size, and orientation.

Equation Simplification

Equation simplification is crucial to understand what type of conic section you are dealing with and to make graphing manageable. A simplified equation lets you easily identify characteristics of the conic, such as its axis alignment and whether it opens vertically or horizontally.To simplify:

• Move all terms to one side of the equation.
• Use rotation transformations, as mentioned before, to eliminate \( xy \)-terms if needed.
• Convert the equation into its standard conic form if possible, such as:
  • Ellipse: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)
  • Parabola: \( (y-k)^2 = 4p(x-h) \) or \( (x-h)^2 = 4p(y-k) \)
  • Hyperbola: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \)

Understanding the equation's structure helps identify the conic's nature and assists in graphing and interpreting its geometric properties efficiently.