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

Eccentricity is 1; the conic is a parabola.

Step 1: Recognize the Conic Type

Observe that the polar equation of the form \( r = \frac{ed}{1 - e \cos \theta} \) represents a conic section where \( e \) is the eccentricity. In this exercise, \( r = \frac{2}{1 - \cos \theta} \), which matches the form with \( ed = 2 \) and \( e = 1 \). Hence, the equation has eccentricity \( e = 1 \).

Step 2: Calculate Eccentricity

From the equation \( r = \frac{2}{1 - \cos \theta} \), we see \( ed = 2 \) and \( e = 1 \). So, the eccentricity \( e \) is 1, which indicates that this conic is a parabola.

Step 3: Identify and Define Conic Type

Since the eccentricity of the conic is \( e = 1 \), this defines a conic section known as a parabola. A parabola is a specific type of conic with an eccentricity of exactly 1.

Step 4: Sketch the Parabola

In polar coordinates, a parabola described by \( r = \frac{2}{1 - \cos \theta} \) opens to the right. To sketch, note that its vertex is located at the pole (\( r = 0 \)), when \( \theta = 0\). The directrix is a vertical line to the left of the pole, and the focus is at the pole itself.

Step 5: Label the Vertices on the Sketch

Since this is a polar form of a conic and specifically a parabola, the vertex for \( r = \frac{2}{1 - \cos \theta} \) is at the origin (pole), as it opens to the right. On the sketch, label the vertex at the pole, with the understanding it lies at the minimum radius point.

Key Concepts

Understanding Eccentricity

Eccentricity is a key concept in understanding different types of conic sections. It is denoted by the letter \( e \) and essentially measures how much a conic section deviates from being circular. The value of eccentricity dictates the type of conic:

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), the conic is an ellipse.
• If \( e = 1 \), the conic is a parabola.
• If \( e > 1 \), the conic is a hyperbola.

In the given problem, the polar equation is \( r = \frac{2}{1 - \cos \theta} \). By comparing it with the general conic formula \( r = \frac{ed}{1 - e\cos\theta} \), we find \( e = 1 \), confirming that it represents a parabola. The eccentricity helps us quickly identify and differentiate conic sections based on these key numerical values.
Understanding eccentricity not only helps in identifying conic types but also plays a role in sketching them correctly.

The Role of Polar Coordinates

Polar coordinates are an alternative to the traditional Cartesian coordinate system. They present points in the plane using a distance from a reference point and an angle from a reference direction. This system is particularly useful when dealing with curves like circles and conics. Here's how it works:

• Instead of \( (x, y) \) in Cartesian coordinates, we use \( (r, \theta) \) where \( r \) is the radius or distance from the origin, and \( \theta \) is the angle measured from the positive x-axis.
• This approach simplifies equations for many types of curves, especially in capturing symmetry and rotations.

In the formula \( r = \frac{2}{1 - \cos \theta} \), the conic is expressed in polar form. This form efficiently describes conics that are centrally symmetric or have radial patterns, like the parabola in this problem. Polar coordinates offer a clear perspective for understanding and plotting such curves.

Exploring the Parabola

A parabola is a type of conic section with a distinctive curve. It is defined geometrically as the set of points equidistant from a point called the focus and a line called the directrix. Key characteristics of a parabola include:

• Its shape resembles a smooth, symmetric U-shaped curve.
• Each parabola is defined by its vertex, the point where it makes its sharpest turn.
• The distance from any point on the parabola to the focus is equal to the perpendicular distance from that point to the directrix.

In our exercise, the equation \( r = \frac{2}{1 - \cos \theta} \) in polar form describes a parabola. This specific parabola opens to the right and has its vertex at the pole, \( \theta = 0 \), which means it lies at the origin in the polar coordinate system. Understanding these properties helps in sketching the parabola accurately and recognizing its orientation. Parabolas are not only mathematically important but also appear in real-world applications like satellite dishes and bridges.