Problem 19 - Precalculus: Mathematics for Calculus

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 xy-term. (c) Sketch the graph. $$11 x^{2}-24 x y+4 y^{2}+20=0$$

Step-by-Step Solution

Answer

It's a hyperbola after eliminating the xy-term.

Step 1: Identify the Form and Coefficients

The given quadratic 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 \( F = 20 \). This is a general second-degree equation or conic section.

Step 2: Calculate the Discriminant

The discriminant \( \Delta \) for a conic section is given by:\[ \Delta = B^2 - 4AC \]Substitute the values:\[ \Delta = (-24)^2 - 4 \times 11 \times 4 = 576 - 176 = 400 \]

Step 3: Determine the Type of Conic Section

With\( \Delta = 400 \), we note that it is greater than zero \( \Delta > 0 \), indicating that the conic section is a hyperbola.

Step 4: Use Rotation to Eliminate xy-Term

To eliminate the \( xy \)-term, we use rotation formulas where:\[ tan(2\theta) = \frac{B}{A-C} = \frac{-24}{11-4} = -\frac{24}{7} \]Solve for \( \theta \).Once \( \theta \) is determined, apply the rotation formulas:\[ x = x'\cos(\theta) - y'\sin(\theta) \]\[ y = x'\sin(\theta) + y'\cos(\theta) \]The goal is to express terms in \( x' \) and \( y' \) without \( x'y' \).

Step 5: Simplify and Sketch the Graph

After rotation and substituting back, the equation should reduce to a standard hyperbola equation. Instead of graphing immediately, simplify to fit the standard form \( \frac{x'^2}{a^2} - \frac{y'^2}{b^2} = 1 \) or \( \frac{y'^2}{b^2} - \frac{x'^2}{a^2} = 1 \). Sketch the hyperbola using the center, vertices, and asymptotes.

Key Concepts

Understanding the Discriminant in Conic Sections

The discriminant is a powerful tool in determining the nature of a conic section from a general quadratic equation of the form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). For determining whether the conic section is an ellipse, hyperbola, or parabola, we use the discriminant \( \Delta \), defined as:\[ \Delta = B^2 - 4AC \]

• If \( \Delta > 0 \), the conic is a hyperbola.
• If \( \Delta = 0 \), the conic is a parabola.
• If \( \Delta < 0 \), the conic is an ellipse.

In the exercise, after calculating: \( \Delta = 400 \), it was concluded that the conic section is a hyperbola because the discriminant is greater than zero. This classification is essential for graphing and understanding the behavior of the conic section.

The Role of Rotation of Axes in Conic Sections

The rotation of axes is a technique used to simplify the form of conic sections by eliminating the \( xy \)-term. Often, the presence of an \( xy \)-term complicates the process of identifying and sketching a conic section.The approach involves using a specific angle \( \theta \), which is found using:\[ \tan(2\theta) = \frac{B}{A-C} \]Applying the formula:\( \tan(2\theta) = \frac{-24}{11-4} = -\frac{24}{7} \).Once \( \theta \) is found:

• Substitute the rotation formulas for \( x' \) and \( y' \).
• Rewrite the original equation in terms of \( x' \) and \( y' \) without the \( x'y' \) term.

This process simplifies the equation, making it easier to identify and graph the conic section’s standard form.

Delving into Hyperbolas

A hyperbola is a type of conic section represented by two open curves called branches. Distinctive for having a negative discriminant value greater than zero, hyperbolas exhibit properties that differ from other conic sections like ellipses and parabolas.Identifying their signature equation, hyperbolas typically appear as:\[ \frac{x'^2}{a^2} - \frac{y'^2}{b^2} = 1 \] or \[ \frac{y'^2}{b^2} - \frac{x'^2}{a^2} = 1 \]Key characteristics of hyperbolas include:

• Two separate branches or curves.
• Asymptotes that guide the curve’s direction.
• A fundamental axis or transverse axis that defines its shape.

Utilizing rotation techniques, the equation from the exercise simplifies into a recognizable hyperbola form, enabling easy sketching using vertices, centers, and asymptotes.

Understanding Ellipses

While the exercise deals with a hyperbola, understanding ellipses is important for grasping conic sections as a whole. An ellipse arises when the discriminant \( \Delta < 0 \). It represents a closed curve, resembling a stretched circle. Ellipses are characterized by:

• A central point, known as the center.
• Two axes: the longer, major axis and the shorter, minor axis.
• Foci, points within the ellipse indicating its shape.

The general equation for an ellipse without a rotation is:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]Understanding these elements is crucial if the discriminant had been less than zero, leading the given conic to be classified as an ellipse.

Characteristics of Parabolas

A parabola is another type of conic section that arises when the discriminant \( \Delta = 0 \). Parabolas open as a single curved path, either towards or away from a focal point. They are distinguished by:

• A vertex, the turning point of the curve.
• An axis of symmetry that divides the parabola into two mirror-image sections.
• A focus, a point around which the parabola is shaped.

The standard equation of a parabola generally appears as:\[ y = ax^2 + bx + c \] or \[ x = ay^2 + by + c \]Despite the original exercise not resulting in a parabola, understanding their properties highlights their significance within conic sections. Each conic section shows unique properties, driven by discriminant values and equation manipulation techniques like rotation.