Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{3}{-4-8 \cos \theta}$$

Step-by-Step Solution

Answer

The equation represents a horizontal ellipse. This is confirmed by graphing the equation which shows the typical oval shape of an ellipse oriented horizontally.

Step 1: Identify the polar equation

The provided polar equation is \( r = \frac{3}{-4-8 \cos \theta} \). We need to identify which type of conic section this equation represents.

Step 2: Rewrite the equation in standard form

Start by simplifying the denominator, yielding \( r = \frac{3}{-4(1+2 \cos \theta)} \). Multiply both sides by the denominator to isolate \( r \), which results in \( r(-1-2\cos \theta) = -3 \). Dividing all terms by -3 gives \(r(1+2\cos \theta) = 1 \). Thus, the standard form of the equation is \( r = \frac{1}{1+2\cos \theta} \), which is the form of a conic section with the focus at the origin, eccentricity e=2 and horizontally oriented directrix. So, the given equation represents a horizontal ellipse.

Step 3: Graph the function

To confirm the result, we can graph the function using a graphing utility. On the graph, it shows that the graph is an ellipse oriented horizontally, which confirms the result.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane at different angles. There are four basic types of conic sections:

• Circles: These are formed when the intersecting plane is perpendicular to the cone's axis.
• Ellipses: These occur when the plane intersects the cone at an angle less than that of the cone, but not perpendicular, creating an oval shape.
• Parabolas: A parabola is formed when the plane is parallel to one of the slopes of the cone.
• Hyperbolas: These arise when the plane intersects both nappes (the upper and lower cones), creating a shape with two separate curves.

Each conic section has unique properties and equations that govern their shapes. For example, in the exercise, we determined that the equation given represents an ellipse due to the specific structure of the equation in polar form. In general, by understanding the characteristics of these conics, students can visualize and analyze their properties more readily.

Standard Form of Conic Equations

The standard form of conic equations is essential for identifying the type of conic section an equation represents. In their most simplified versions, the standard forms make the properties of conics—such as the center, the vertices, and the foci—clear and easy to determine.
For a polar equation of a conic with a focus at the origin, like the one in our exercise, the standard form is given by:\[\begin{equation} r = \frac{e}{1 + e\cos \theta}text{ or }\text{ in case of a vertical directrix }\frac{e}{1 + e\sin \theta}text{.}text{}text{}text{}text{}text{}text{}text{}text{}text{}text{}\end{equation}\]in which \( e \) represents the eccentricity. If \( e < 1 \), it's an ellipse; \( e = 1 \) denotes a parabola; and \( e > 1 \) signifies a hyperbola. By comparing the given equation to the standard forms, students can identify conic sections just by looking at the equation's structure. In our exercise, rearranging the equation \( r = \frac{3}{-4-8 \cos \theta} \) into standard form made it clear that we're dealing with a conic section that is an ellipse, as the eccentricity (e) value is greater than one.

Graphing Utility

A graphing utility is an invaluable tool for confirming the type of conic section represented by an equation. These utilities allow students to visualize the equation's graph without having to plot many points manually. This visualization can help reinforce the identification of the conic through its physical shape.
In our example, after identifying the equation as representing an ellipse, we use a graphing utility to plot the polar equation. By entering the simplified equation, the graphing utility provides a graph that aligns with the characteristics of an ellipse, namely its symmetrical, oval shape. This confirms our earlier analysis and helps in understanding the correlation between the mathematical equation and its graphical representation.
Graphing utilities may also offer additional features like zooming, changing the scale, and tracing points, making it easier to study the curvature and aspects like the major and minor axes, which are critical for a thorough understanding of conic sections.