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Chapter 11: Problem 20

Analyze each equation and graph it. \(r=\frac{8}{2+4 \cos \theta}\)

Short Answer

Expert verified
The equation \( r=\frac{8}{2+4 \cos \theta} \) represents a hyperbola. Graph it by plotting points based on its properties and reflecting them across the directrix.

Step by step solution

01

Identify the Type of Equation

The given equation is in polar form: \( r = \frac{8}{2 + 4 \cos \theta} \). It represents a conic section with one focus at the origin.
02

Simplify the Equation

Simplify the equation by factoring the denominator: \( r = \frac{8}{2 + 4 \cos \theta} = \frac{8}{2 (1 + 2 \cos \theta)} = \frac{4}{1 + 2 \cos \theta} \).
03

Identify the Conic Section

Compare the simplified equation \( r = \frac{4}{1 + 2 \cos \theta} \) to the standard form of the polar equation for a conic section: \( r = \frac{ed}{1 + e \cos \theta} \). Here, \( e = 2 \) and \( ed = 4 \), so \( d = 2 \).
04

Determine the Nature of the Conic Section

Given \( e = 2 \), which is greater than 1, the conic section is a hyperbola.
05

Graph the Equation

To graph the hyperbola, note the properties identified: the focus is at the origin, and the directrix is at \( r = 2 \). Plot the points based on the given equation and reflect them across the directrix.
06

Label Key Features on the Graph

Mark the focus at the origin, the vertices, and the asymptotes. The vertices can be plotted where \( \theta = 0 \) and \( \theta = \pi \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conic Sections
Conic sections are shapes created when a plane intersects a cone. There are four types of conic sections: circles, ellipses, parabolas, and hyperbolas. They describe many of the possible paths of objects moving under the influence of gravity, such as planets and comets. Each conic section has a specific set of properties that distinguish it from the others. Understanding these properties is essential in graphing and analyzing their equations.
Hyperbolas
A hyperbola is one of the conic sections that occur when a plane cuts through both nappes (the upper and lower parts) of a cone. Hyperbolas consist of two separate curves called branches. They are defined by their two foci and asymptotes. The standard form of the hyperbola's polar equation is \( r = \frac{ed}{1 + e \cos \theta}\). When the eccentricity (\(e\)) is greater than 1, the conic section is a hyperbola. The distance between the two branches increases as they extend further from the center.
Graphing Polar Coordinates
Graphing in polar coordinates involves representing equations with radii and angles, instead of x and y coordinates. Each point on a polar graph is determined by an angle \( \theta\) and a radius \( r \), originating from a central point known as the pole. To graph the equation \( r = \frac{4}{1 + 2 \cos \theta}\), you start by identifying key points, such as vertices and foci. Reflecting points across the directrix helps in plotting the correct shape of the graph systematically. Remember to label the key features, such as the focus and vertices, for clarity.
Rational Functions
Rational functions are ratios of polynomial expressions. They can be written in the form \( \frac{P(x)}{Q(x)} \), where \ P(x) \ and \ Q(x) \ are polynomials, and \ Q(x) \ is not equal to zero. In the context of polar equations for conic sections, rational functions help to describe the relationship between the radius \( r \) and the angle \( \theta \). Simplifying these ratios aids in identifying the type of conic section and its properties. Here, \ r = \frac{4}{1 + 2 \cos \theta} \ is a rational function describing a hyperbola in polar form.

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Most popular questions from this chapter

Find the center, transverse axis, vertices, foci, and asymptotes, Graph each equation. \(\frac{(y+3)^{2}}{4}-\frac{(x-2)^{2}}{9}=1\)

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Use the fact that the orbit of a planet about the Sun is an ellipse, with the Sun at one focus. The aphelion of a planet is its greatest distance from the Sun, and the perihelion is its shortest distance. The mean distance of a planet from the Sun is the length of the semimajor axis of the elliptical orbit. A planet orbits a star in an elliptical orbit with the star located at one focus. The perihelion of the planet is 5 million miles. The eccentricity \(e\) of a conic section is \(e=\frac{c}{a} .\) If the eccentricity of the orbit is \(0.75,\) find the aphelion of the planet.
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