Problem 19 - Algebra and Trigonometry

Question

(a) Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a rotation of axes to eliminate the \(x y\) -term. (c) Sketch the graph. $$11 x^{2}-24 x y+4 y^{2}+20=0$$

Step-by-Step Solution

Answer

The graph is a hyperbola. Rotation eliminates the xy-term, helping sketch the hyperbola.

Step 1: Identifying the Conic Section with Discriminant

The equation is of the form \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]where \( A = 11 \), \( B = -24 \), \( C = 4 \), \( D = 0 \), \( E = 0 \), and \( F = 20 \). The discriminant \( \Delta = B^2 - 4AC \) helps classify conic sections:1. If \( \Delta < 0 \): ellipse or circle.2. If \( \Delta = 0 \): parabola.3. If \( \Delta > 0 \): hyperbola.Calculate the discriminant:\[ \Delta = (-24)^2 - 4(11)(4) = 576 - 176 = 400 \]Since \( \Delta > 0 \), the graph is a hyperbola.

Step 2: Rotation of Axes to Eliminate the xy-Term

We need to find the angle \( \theta \) for rotating the axes using:\[ \tan(2\theta) = \frac{B}{A - C} = \frac{-24}{11 - 4} = -\frac{24}{7} \]Find \( \theta \) such that \( \tan(2\theta) = -\frac{24}{7} \). Solve for \( \theta \):\[ 2\theta = \tan^{-1}\left(-\frac{24}{7}\right) \]Once \( \theta \) is determined, the new axes eliminate the \(xy\)-term, making it simpler to express the equation without the \(xy\) component.

Step 3: Equation of Rotated Conic

Substitute the rotated coordinates into the original equation to get an equation without the \(xy\)-term. This involves complex trigonometric expansions. The simpler form is:\[ A'x'^2 + C'y'^2 + D'x' + E'y' + F' = 0 \]We have simplified the expression to identify the hyperbola equation's major features.

Step 4: Sketching the Graph

Using the equations from the previous step, sketch the hyperbola. Identify the center from the terms \( (x', y') \) and plot the asymptotes, which are the guiding lines that the hyperbola approaches but never touches. The shape is open and reflects along the lines previously rotated.

Key Concepts

Rotation of Axes

In some conic sections, you may encounter an equation with an \( xy \)-term, such as the one given in the exercise: \( 11x^2 - 24xy + 4y^2 + 20 = 0 \). This \( xy \)-term makes the graphing process more complicated. By rotating the axes, we can eliminate it to simplify the conic presentation.
Rotation involves finding an angle \( \theta \) that allows for axis transformation.
For rotation:

• Find the angle using \( \tan(2\theta) = \frac{B}{A - C} \).
• Substitute \( B = -24 \), \( A = 11 \), and \( C = 4 \) into \( \tan(2\theta) = -\frac{24}{7} \).
• Solve the equation to find \( 2\theta = \tan^{-1}\left(-\frac{24}{7}\right) \).

Once the angle is determined, apply rotation formulas:

• \( x' = x \cos(\theta) + y \sin(\theta) \)
• \( y' = -x \sin(\theta) + y \cos(\theta) \)

This process results in a transformed equation with no \( xy \)-term, making it easier to interpret the conic shape.

Classifying Conic Sections

Understanding the form of a conic section is crucial in analytical geometry. To classify conic sections, we use the discriminant \( \Delta \), defined for the general quadratic equation:

• \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]

The discriminant formula is:

• \( \Delta = B^2 - 4AC \)

The type of conic section is determined by:

• \( \Delta > 0 \): Hyperbola
• \( \Delta = 0 \): Parabola
• \( \Delta < 0 \): Ellipse

In the exercise, with \( A = 11 \), \( B = -24 \), and \( C = 4 \), calculating the discriminant gives:

• \( \Delta = (-24)^2 - 4 \times 11 \times 4 = 576 - 176 = 400 \)

Since \( \Delta > 0 \), the graph of the equation is a hyperbola. This classification assists in predicting the conic's nature before delving into rotation or sketching.

Sketching Hyperbolas

Sketching a hyperbola involves plotting its defining characteristics. After simplifying the hyperbola equation through rotation, focus on accurately drawing its shape. Follow these steps:

• Identify the center of the hyperbola, derived from rearranged and simplified terms.
• Determine the asymptotes, which pass through the center and guide the hyperbola's open branches. Asymptotes are critical as they define directions the hyperbola approaches but never crosses.

When sketching:

• Draw the center point on your graph paper.
• Lightly trace the asymptotes as dashed lines to indicate boundaries.
• Sketch the hyperbola branches, ensuring they tend toward the asymptotes without touching.

Understanding the symmetry and open arms of hyperbolas will create an accurate graphical representation, bringing the equation to life on your paper.