Problem 29 - 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. \(r=\frac{4}{1+\cos \theta}\)

Step-by-Step Solution

Answer

The curve is a parabola with eccentricity 1.

Step 1: Identify the Equation Type

The given equation is in the form \( r = \frac{ed}{1 + e\cos\theta} \), which represents a conic section. Specifically, this form resembles the equation of a conic with respect to the polar coordinate system, where \( e \) is the eccentricity, and \( d \) is a constant.

Step 2: Identify the Eccentricity

In the form \( r = \frac{ed}{1 + e\cos\theta} \), compare it with the given equation \( r = \frac{4}{1 + \cos\theta} \). We find that \( ed = 4 \) and \( e = 1 \). Thus, the eccentricity \( e \) is 1.

Step 3: Classify the Conic Section

Since the eccentricity \( e = 1 \), the conic section is a parabola. In polar coordinates, an equation with \( e = 1 \) corresponds to a parabola.

Step 4: Sketch the Graph

For \( r = \frac{4}{1 + \cos \theta} \), the polar graph will depict a parabola with its focus at the origin. The directrix is vertically oriented, due to the \( \cos \theta \) term, and positioned at \( x = -4 \) from the polar system's pole (the origin).

Key Concepts

Conic Sections

In mathematics, conic sections are curves obtained by intersecting a cone with a plane. They are crucial in understanding the properties of geometric shapes, particularly in polar coordinates.
Conic sections include:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

These curves are defined by their distance from a fixed point (focus) and a fixed line (directrix). The position and shape depend on the eccentricity, which we’ll discuss shortly.
Understanding conic sections can help us solve problems in physics, engineering, and astronomy.

Eccentricity

Eccentricity is a key parameter in determining the shape of a conic section. Denoted by the symbol \( e \), it's a measure of how much a conic section deviates from being circular.
Here's how eccentricity relates to each conic:

• Circle: \( e = 0 \)
• Ellipse: \( 0 < e < 1 \)
• Parabola: \( e = 1 \)
• Hyperbola: \( e > 1 \)

In the exercise, the given polar equation is \( r = \frac{4}{1 + \cos \theta} \). Comparing it with the general conic equation in polar form, we find \( e = 1 \), identifying the conic as a parabola.
Eccentricity helps describe the conic’s behavior and the relative positioning of its components in space.

Parabola Graph

A parabola is a particular conic section that results when the plane is parallel to the cone's edge. It can open up, down, left, or right, depending on the orientation of the equation.
In polar coordinates, a parabola with eccentricity \( e = 1 \) is symmetric about a line, and its focus is at the origin. The directrix, for this exercise, is vertical, given by the term containing \( \cos \theta \).
Key features of a parabola include:

• Vertex: The point where the parabola changes direction.
• Axis of symmetry: The line that divides the parabola into mirror images.
• Focus: A point inside the parabola used to define the curve.
• Directrix: A line outside the parabola that works with the focus to set its shape.

The graph of \( r = \frac{4}{1 + \cos \theta} \) shows a parabola with its vertex not at the pole, but displaced along the directrix.

Polar Equation

Polar equations represent curves on a plane, different from the usual Cartesian coordinates. They express the relationship between the radius \( r \) and the angle \( \theta \), originating from the pole often considered as the origin.
The general form of a conic section's polar equation is \( r = \frac{ed}{1 + e\cos \theta} \), where \( e \) is eccentricity and \( d \) is a constant. This allows for a straightforward classification of conics based on eccentricity.
In polar equations:

• A positive \( \cos \theta \) term positions the conic horizontally.
• A positive \( \sin \theta \) term indicates a vertical alignment.
• The denominator's form determines the overall symmetry.

The polar equation \( r = \frac{4}{1 + \cos \theta} \) corresponds to a parabola, emphasizing its symmetric properties and focus orientation in the polar system.