Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=\frac{4}{2+2 \cos (\theta-\pi / 3)}\)

Step-by-Step Solution

Answer

The curve is a parabola with eccentricity 1, symmetric about \(\theta = \pi/3\).

Step 1: Recognize the Polar Equation Form

The given polar equation is \(r = \frac{4}{2 + 2 \cos (\theta - \pi / 3)}\). Compare this to the general conic section form \(r = \frac{ed}{1 + e \cos(\theta - \theta_0)}\). To make the forms comparable, we can rewrite the given equation as \(r = \frac{2}{1 + \cos(\theta - \pi / 3)}\). The equation is in the form of a conic section with potential eccentricity \(e\) and is related to the cosine function.

Step 2: Identify the Type of Conic

In the rewritten equation \(r = \frac{2}{1 + \cos(\theta - \pi / 3)}\), the denominator is \(1 + e \cos(\theta - \theta_0)\), indicating a conic centered at \(\theta_0 = \pi/3\). The constant in front of the cosmic term is compared to "1 + e" from the general form. Therefore, the equation is that of an ellipse since \(e = 1\).

Step 3: Calculate the Eccentricity e

The form of the equation suggests \(1 + e = 2\), giving \(e = 1\). Therefore, the eccentricity \(e\) for this conic is \(0\), indicating a circular conic, but because polar format is used in traditional conic form when \(e=1\), it identifies a parabola whose eccentricity is considered as 1 for standard understanding.

Step 4: Sketch the Graph

Since this is the polar equation of a parabola (with \(e = 1\)), it opens depending on the reference angle \(\theta - \theta_0\). It is symmetric about the line \(\theta = \pi/3\) and its vertex can be identified in the polar plot by plugging in \(\theta = \pi/3\) leading to center-based parabolic graph pointing outward.

Key Concepts

Conic Sections

Conic sections are shapes created by intersecting a plane with a cone. The angle and position at which this intersection occurs determine the type of section formed.
There are four main types of conic sections:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

In polar coordinates, these sections can be described using equations that highlight their unique properties.
These polar equations generally take the form: \( r = \frac{ed}{1 + e \cos(\theta - \theta_0)} \) or \( r = \frac{ed}{1 + e \sin(\theta - \theta_0)} \). These general forms help identify the type of conic section based on the value of the eccentricity \(e\). Understanding these details assists in distinguishing between the different conic sections.

Eccentricity

Eccentricity is a measure that describes how much a conic section deviates from being circular. It's a crucial parameter that helps in identifying the specific type of conic section.
Eccentricity, denoted as \(e\), helps classify:

• A circle where \(e = 0\)
• An ellipse where \(0 < e < 1\)
• A parabola where \(e = 1\)
• A hyperbola where \(e > 1\)

Recognizing the value of \(e\) allows us to determine whether the equation represents a circle, an ellipse, a parabola, or a hyperbola.
For example, in our exercise, identifying the eccentricity as \(e = 1\) helps confirm the conic as a parabola.

Ellipse

An ellipse is a curved shape that looks like a stretched circle. It has two focal points, and for any point on the ellipse, the sum of the distances to these focal points is constant.
When dealing with polar equations, if the eccentricity \(e\) is greater than 0 but less than 1, the conic section is an ellipse.
Ellipses have unique properties like:

• Two axes: a major axis (the longest diameter) and a minor axis (the shortest diameter).
• The eccentricity is less than 1, describing how much it is stretched out compared to a circle.

In coordinate geometry, ellipses are typically represented with the equation \((x^2/a^2) + (y^2/b^2) = 1\), but in polar form, identifying \(0 < e < 1\) as eccentricity confirms the elliptical nature.

Parabola

Parabolas are unique conic sections that have one focal point and one directrix. When the coefficient of the trigonometric term in a polar equation results in \(e = 1\), the conic section represents a parabola. This is significant in understanding how conics behave in polar plots.
Parabolas have some interesting properties, such as:

• A single axis of symmetry.
• The distance from any point on the parabola to the focus equals the distance to the directrix.

In the exercise discussed, the polar equation given transforms into that of a parabola when compared to its standard form, with the eccentricity revealing its parabolic nature.
This helps in accurately sketching its graph, with the parabola opening depending on changes in the angle \( \theta \) relative to the reference direction.