Problem 31 - Algebra and Trigonometry

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 eccentricity is 1, identifying the conic as a parabola.

Step 1: Identify Conic Equation Form

The given equation is in the polar form \( r = \frac{ed}{1 - e\cos(\theta)} \), where \( e \) is the eccentricity.

Step 2: Determine Eccentricity

In the equation, compare with \( r = \frac{ed}{1 - e\cos(\theta)} \). We see it in the form \( r = \frac{T}{1 - \cos(\theta)} \), indicating \( e = 1 \).

Step 3: Identify Conic Type

Since \( e = 1 \), the conic is a parabola. Eccentricity \( e \) equals 1 characterizes a parabola.

Step 4: Sketch the Conic

To sketch the conic, plot the parabola's vertex at the origin in the polar coordinate system. The focus is at \((1, 0)\) on the positive x-axis.

Step 5: Label the Vertices

In a parabola, the vertex is at the origin \((0, 0)\) in this polar setup. Label this point on the sketch.

Key Concepts

Eccentricity

Eccentricity is a fundamental concept when exploring conic sections, which include ellipses, parabolas, and hyperbolas. It is a number that describes how "stretched" or "flattened" a conic section is. For each type of conic:

• An ellipse has an eccentricity less than 1.
• A parabola has an eccentricity equal to 1.
• A hyperbola has an eccentricity greater than 1.

In the given exercise, we have the polar equation \[ r = \frac{2}{1-\cos \theta} \]This equation is of the form \[ r = \frac{ed}{1-e\cos(\theta)} \] which allows us to determine the eccentricity \( e \).
On comparing, the equation tells us that \( e = 1 \), indicating that the conic is a parabola. The understanding of eccentricity helps in classifying conic sections based on their shape and mathematical properties.

Polar Coordinates

Polar coordinates provide a way of locating a point in a plane based on its distance from a reference point, often called the origin, and its angle from a reference direction, which is usually the positive x-axis. In contrast to Cartesian coordinates that use \((x, y)\), polar coordinates are denoted as \((r, \theta)\).
Let's break it down:

• \(r\) represents the radial distance from the origin.
• \(\theta\) is the angle from the positive x-axis in a counter-clockwise direction.

In the problem, the polar equation \[ r = \frac{2}{1 - \cos \theta} \] provides all information needed in terms of distance \( r \) and angle \( \theta \). Polar coordinates simplify the analysis of curves that exhibit rotational symmetry, like circles and the derived conic sections. They offer a unique perspective on geometry, emphasizing angular relationships and radial distances.

Parabola

A parabola is a specific type of conic section that exhibits a unique, symmetric U-shape. In algebraic terms, a parabola is defined as the set of points equidistant from a fixed point (the focus) and a line (the directrix). In the case of the polar equation addressed in this exercise, \[ r = \frac{2}{1 - \cos \theta} \]we derived that this equation represents a parabola. The characteristics of a parabola include:

• The vertex, where the parabola changes direction, which in polar coordinates is at the origin \((0,0)\).
• The axis of symmetry, a line that runs directly through the vertex and focus.

In this context, the equation's conic traits reveal that the focus is positioned at \((1, 0)\) on the positive x-axis. The polar setup helps vividly sketch and visualize the parabola, showing its balance and structure at the origin. Understanding parabolas in this form highlights their geometric elegance and mathematical consistency.