Problem 12 - College Algebra

Question

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{2}{1-\cos \theta} $$

Step-by-Step Solution

Answer

The equation represents a parabola with eccentricity 1 and directrix \( x = 2 \).

Step 1: Identify the Form of the Equation

The given polar equation is \( r = \frac{2}{1 - \cos \theta} \). This matches the form \( r = \frac{ed}{1 - e\cos \theta} \), which represents a conic section with the focus at the origin. In this form, \( e \) is the eccentricity and \( d \) is the directrix.

Step 2: Determine the Eccentricity

Compare the given equation \( r = \frac{2}{1 - \cos \theta} \) with the standard form \( r = \frac{ed}{1 - e\cos \theta} \). Notice that \( ed = 2 \) and \( e = 1 \). Since the eccentricity \( e = 1 \), this conic is a parabola.

Step 3: Find the Directrix

Since the eccentricity \( e = 1 \), we can determine the directrix by using \( ed = 2 \). Substituting \( e = 1 \), we find \( d = 2 \). Thus, the directrix for this parabola is the line \( x = 2 \).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a two-dimensional plane through two values: a distance from a reference point (often called the origin) and an angle from a reference direction. Unlike the regular Cartesian coordinates, which use x and y to indicate position, polar coordinates use

• the radial coordinate (r), giving the distance from the origin, and
• the angular coordinate (θ), indicating the angle measured from a reference direction, typically the positive x-axis.

This system is often helpful in contexts where the position is naturally circular or radial, like in problems involving conic sections and their equations in polar form. For instance, using polar coordinates, we can write the equation of a conic section centered at the origin in terms of r and θ, which might make some problems easier to solve due to their symmetry.

Eccentricity

Eccentricity is a parameter that defines the shape of a conic section. It’s a number that determines how much a conic section deviates from being circular.

• If the eccentricity (\( e \)) is equal to 0, the conic is a circle.
• When \( e \) is between 0 and 1, it forms an ellipse.
• If \( e \) equals 1, the conic is a parabola.
• For values greater than 1, the conic is a hyperbola.

In the context of the problem, the eccentricity was found to be 1, signifying that the conic is a parabola. This concept not only helps identify the type but also gives insight into the characteristics of the conic itself. Eccentricities are crucial when dealing with polar coordinates because they are directly tied into the equations represented in these systems.

Directrix

The directrix is a fixed line used in the description of a conic section. For each conic section, there is a specific definition:

• In a circle, the concept of a directrix isn't used.
• For ellipses and hyperbolas, there are typically two directrices.
• In a parabola, however, there is only one directrix.

This line is related to the focus of the conic, and it's used along with the eccentricity to determine the other characteristics of the shape.
A parabola's directrix is key to its reflective properties and geometric definition. In the case of polar coordinates, the equation contains a term 'ed', where 'd' is the distance from the origin to the directrix. From our problem, we know that when the eccentricity (\( e \)) is 1, the directrix can be calculated easily. Here the directrix was found to be at the line \( x = 2 \).

Parabola

A parabola is a type of conic section that has some unique properties. When described in polar coordinates, it always has an eccentricity (\( e \)) of 1. This distinctive feature makes it easy to identify from its equations.
Parabolas are intriguing as they have a single focus and a single directrix. What's fascinating about parabolas is their geometric property where each point is equidistant from the focus and the directrix line.
This reflective property is famously used in satellite dishes and car headlights to focus signals and light. In the given problem, the parabola was shown through an equation in polar form: \( r = \frac{2}{1 - \cos \theta} \). This is how we naturally deduced that the conic is a parabola. Recognizing the parabola's properties is essential in many fields, including physics and engineering, due to its practical applicability.