Problem 33 - 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. $$ r=\frac{4}{2+2 \cos \theta} $$

Step-by-Step Solution

Answer

The curve is a parabola with eccentricity 1.

Step 1: Identify the Form of the Polar Equation

The given polar equation is \( r = \frac{4}{2 + 2 \cos \theta} \). Consider the general form for conic sections in polar coordinates: \( r = \frac{ed}{1 + e\cos \theta} \), where \(e\) is the eccentricity and the conic is centered on the polar axis. We'll manipulate the equation to see if it matches this form.

Step 2: Simplify and Compare with Conic Equation

Simplify the given equation: \( r = \frac{4}{2 + 2 \cos \theta} = \frac{2}{1 + \cos \theta} \). This is now in the form \( r = \frac{ed}{1 + e\cos \theta} \) with \( ed = 2 \) and \( e = 1 \), considering the denominator adjustments.

Step 3: Identify the Conic Type and Eccentricity

Since \( e = 1 \), this characterizes the conic as a parabola. In conic sections, \( e = 1 \) indicates a parabola with directrix symmetry along the polar axis.

Step 4: Sketch the Parabola Using Polar Coordinates

For \( r = \frac{2}{1 + \cos \theta} \), as \( \theta \) varies, the curve forms a parabola with its vertex at the pole (origin). The parabola opens to the left since it results from \( 1 + e\cos \theta \), with focus on the axis extending to negative direction of \( \theta = 0 \). Plot several points for \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) to better sketch the shape.

Key Concepts

Conic Sections

Conic sections are a fascinating family of curves that arise when a plane intersects a double-napped cone. These curves include ellipses, parabolas, and hyperbolas. Each type of conic section has its own unique properties and equations, which can be expressed in both Cartesian and polar coordinates.

• Ellipse: The eccentricity for ellipses is less than 1, meaning the curve is a closed loop.
• Parabola: When the eccentricity is exactly 1, the conic is a parabola, which opens either upward, downward, or sideways.
• Hyperbola: The eccentricity for hyperbolas is greater than 1, resulting in two separate curves.

In the context of polar equations, conic sections are described by the formula \( r = \frac{ed}{1 + e\, \cos \theta} \) or \( r = \frac{ed}{1 + e\,\sin\theta} \), where \( e \) is the eccentricity and \( d \) is a constant that determines the location of the directrix. Depending on the value of \( e \), we can classify the curve as an ellipse, parabola, or hyperbola.

Eccentricity

Eccentricity is a crucial component for identifying and characterizing conic sections. It determines how much the conic deviates from being circular. In mathematics, eccentricity, denoted by \( e \), is defined as the ratio of the distance of any point on the conic from the focus to its perpendicular distance from the directrix.

• If \( e = 0 \), it represents a circle, which is a special case of an ellipse.
• If \( 0 < e < 1 \), the conic is an ellipse, demonstrating a closed, oval shape.
• If \( e = 1 \), the conic is identified as a parabola, which is a curve that provides the same distance from the focus and directrix.
• If \( e > 1 \), the conic is a hyperbola, typically displaying two distinct, separate curves.

For the given polar equation \( r = \frac{2}{1 + \cos \, \theta} \), the eccentricity \( e \) values as 1. This precisely categorizes the conic as a parabola.

Parabola

A parabola is a unique type of conic section characterized by its distinctive "U" shape. The standard definition describes a parabola as the set of all points equidistant from a particular point, called the focus, and a line known as the directrix.

• In polar coordinates, parabolas can exhibit different orientations based on the angle variable \( \theta \).
• For example, the equation \( r = \frac{ed}{1 + e\cos \theta} \) will yield a parabola that opens sideways.

In the given equation \( r = \frac{2}{1 + \cos \theta} \), the parabola opens to the left, with the vertex located at the pole (origin). As \( \theta \) varies, points on the curve will trace out a parabola symmetrical about the polar axis, indicating the focus lies along this axis.
When sketching such a parabola, consider plotting points for key angles like \( \theta = 0, \frac{\pi}{2}, \pi, \text{ and } \frac{3\pi}{2} \). These points help illustrate the symmetric nature of the parabolic curve in polar coordinates.