Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=\frac{6}{4-\cos \theta} $$

Step-by-Step Solution

Answer

The curve is a parabola with eccentricity 1.

Step 1: Transform to Standard Form for Conics

We start with the given polar equation \( r = \frac{6}{4 - \cos \theta} \). The standard form for a conic in polar coordinates is \( r = \frac{ed}{1 - e\cos \theta} \), where \( e \) is the eccentricity. To compare both forms, we multiply both sides by \(4 - \cos \theta\) to get \( r(4 - \cos \theta) = 6 \).

Step 2: Simplify the Equation

Simplify the equation by distributing \( r \): \( 4r - r \cos \theta = 6 \), and rearrange it to \( r \cos \theta = 4r - 6 \). Solve for \( r \): \( r = \frac{6}{4 - \cos \theta} \).

Step 3: Identify Conic Parameters

Notice that our equation matches \( r = \frac{ed}{1 - e\cos \theta} \) with \( d = 6 \) and the denominator \( 1 - e \cos \theta \) replaced by \( 4 - \cos \theta \). Hence, \( 4 = 1 \) and \( e = 1 \). Since \( e = 1 \), this is a parabola.

Step 4: Conclusion

Since \( e = 1 \), the conic is a parabola with an eccentricity of 1. Therefore, the curve is a parabola.

Step 5: Sketch the Graph

To sketch the parabola in polar coordinates, note that the parabola will have its axis along the horizontal (\(\theta = 0\) line). The vertex will be at the focus, which in polar form has the directrix at \( r = \frac{6}{4}\) at \( \theta = 0 \), where the curve directly orients away from the pole.

Key Concepts

Conic Sections

Conic sections are curves generated by the intersection of a plane with a double napped cone. These curves include ellipses, parabolas, hyperbolas, and circles. Each type of conic has unique characteristics that are determined by its eccentricity. The polar equations often used to define conics are helpful for visualizing these curves in a coordinate system where angles and distances play central roles.
In polar coordinates, conics take the form \( r = \frac{ed}{1 - e \cos \theta} \), where:

• \( e \) is the eccentricity
• \( d \) is a constant representing the distance from the pole

Depending on the value of \( e \), you can determine the type of conic. If \( e = 0 \), it's a circle; if \( 0 < e < 1 \), it's an ellipse; if \( e = 1 \), it's a parabola; and if \( e > 1 \), it's a hyperbola. Understanding these distinctions is crucial for naming curves in exercises involving polar equations.

Parabola

A parabola is a fascinating conic section that can be described as the set of all points equidistant from a fixed point, called the focus, and a line, called the directrix. In the context of polar equations, a parabola serves as a bridge between different conic sections through its unique property of having an eccentricity \( e = 1 \).

In the equation, \( r = \frac{6}{4 - \cos \theta} \), we identify it as a parabola by realizing that it matches the standard form \( r = \frac{ed}{1 - e \cos \theta} \) with \( e = 1 \). The presence of \( e = 1 \) indicates that the path traced by this equation is a parabola stretching infinitely along the plane. This specific orientation and value of \( e \) make parabolas recognizable and distinguishable from other conic sections.

Eccentricity

Eccentricity is a core concept in understanding conic sections. It defines how "stretched" or "spread out" a conic section is. In simple terms, eccentricity \( e \) measures the deviation of a conic section from being circular.

• For a circle, \( e = 0 \), indicating there's no deviation from center symmetry.
• Ellipses have \( 0 < e < 1 \), suggesting a symmetrical curve but stretched.
• For parabolas, \( e = 1 \), highlighting that these curves are open and extend infinitely.
• Hyperbolas have \( e > 1 \), showing how these curves open away from each other.

In the given equation \( r = \frac{6}{4 - \cos \theta} \), the eccentricity is identified as \( e = 1 \), marking it as a parabola. Recognizing \( e \) allows you to predict a conic section's general shape and behavior within its geometric space.

Graph Sketching

Graph sketching in polar coordinates can seem daunting at first, but with an understanding of the structure of conics, it becomes more intuitive. Sketching a curve based on the equation \( r = \frac{6}{4 - \cos \theta} \) involves recognizing that this represents a parabola.

To sketch this, keep in mind:

• The vertex of the parabola in polar coordinates is at the point closest to the pole.
• The directrix is a vertical line against which distances are measured.
• The graph extends infinitely in the direction indicated by the equation, due to the eccentricity value of 1.

When sketching, focus on plotting key points like where \( \theta = 0 \) and aligning the graph along its axis of symmetry. Format the direction and space of your graph according to these principles, and you can accurately represent the parabola in your coordinate system.