Problem 31 - Precalculus: Mathematics for Calculus

Question

(a) Find the eccentricity, and identify the conic. (b) Sketch the conic, and label the vertices. $$r=\frac{2}{1-\cos \theta}$$

Step-by-Step Solution

Answer

The conic is a parabola with an eccentricity of 1, vertex at (2,0).

Step 1: Identify the Conic Type

The given polar equation is \( r = \frac{2}{1 - \cos \theta} \). This equation is of the form \( r = \frac{ed}{1 - e\cos \theta} \). This form suggests a conic section with the directrix perpendicular to the polar axis. Here, \( e \) is the eccentricity. Identify \( e \) by comparing the formula with the given equation.

Step 2: Calculate the Eccentricity

Compare the given equation \( r = \frac{2}{1 - \cos \theta} \) to the standard polar form \( r = \frac{ed}{1 - e\cos \theta} \). You find that \( e = 1 \) since it matches with \( 1 \cdot d = 2 \).

Step 3: Conic Type Determination

When \( e = 1 \), the conic is a parabola. Thus, the conic described by the equation \( r = \frac{2}{1 - \cos \theta} \) is a parabola.

Step 4: Sketch the Parabola

Since the equation is \( r = \frac{2}{1 - \cos \theta} \), this parabola opens to the right. The pole (origin) is one of the focal points, and the vertex is at \( \theta = 0 \) on the positive x-axis. Plot the parabola on the polar coordinate system, with its vertex at \( (2,0) \).

Step 5: Label the Parabola's Vertex

The vertex of the parabola is at the point where \( \theta = 0 \) in polar coordinates. Thus, in Cartesian coordinates, the vertex is at \( (2,0) \). Label this point as the vertex on the sketch.

Key Concepts

Eccentricity

Eccentricity is a fundamental concept when discussing conic sections. It helps determine the shape of the conic. The eccentricity, often denoted by the symbol \( e \), measures the deviation of the conic from being circular. Each type of conic section - circles, ellipses, parabolas, and hyperbolas - is characterized by its eccentricity:

• For a circle, \( e = 0 \).
• For an ellipse (excluding circles), \( 0 < e < 1 \).
• For a parabola, \( e = 1 \).
• For a hyperbola, \( e > 1 \).

In the problem, we identified the conic type by calculating the eccentricity. The polar equation given was \( r = \frac{2}{1 - \cos \theta} \), which directly follows the form \( r = \frac{ed}{1 - e\cos \theta} \). By comparing the structures, we found that \( e = 1 \). This indicated that the conic is a parabola, as parabolas have an eccentricity of 1.

Parabola

A parabola is a special type of conic section you encounter when the eccentricity \( e \) is exactly 1. Geometrically, a parabola is defined as the set of all points equidistant from a focus point and a directrix (a fixed line).
In polar coordinates, a parabola is usually represented by equations of the form \( r = \frac{ed}{1 - e \cos \theta} \) or \( r = \frac{ed}{1 - e \sin \theta} \). For our given problem, the equation was \( r = \frac{2}{1 - \cos \theta} \).
This form tells us that our parabola opens horizontally, specifically to the right. The vertex of the parabola is at maximum distance from the directrix and is directly aligned with the focus of the parabola. In the sketch, you would plot this parabola on a polar graph with the vertex of \( (2, 0) \) located on the positive x-axis.

Polar Coordinates

Polar coordinates provide a unique way to describe curves and shapes based on angles and distances from a central point or pole, rather than from a fixed Cartesian axis. Every point is defined by \( (r, \theta) \), where \( r \) is the radial coordinate (distance from the pole) and \( \theta \) is the angular coordinate (angle from the positive x-axis).
The given exercise involved analyzing the equation \( r=\frac{2}{1-\cos \theta} \), representing a parabola in polar coordinates. In these coordinates, it's often easier to see symmetry and directional orientation of a conic section, as transformations involve simple changes in \( r \) and \( \theta \).

• When the eccentricity \( e \) is found, the form of the equation helps in determining the type of conic.
• Polar plots can directly show how the conic opens and its orientation based on \( \theta \).
• This flexibility makes analyses of conics, like parabolas, very intuitive in polar systems.

Understanding how to interpret and convert between polar and Cartesian plots enriches comprehension of plane geometry, offering a broader application for real-world and theoretical problems.