Problem 76 - Algebra and Trigonometry

Question

Graph each polar equation. $$ r=\frac{1}{1-\cos \theta} \quad(\text {parabola}) $$

Step-by-Step Solution

Answer

The graph is a parabola extending indefinitely outward for \( \theta = 0\) and forming a curve for other values of \(\theta\).

Step 1: Identify the type of conic

The given equation is in the form of a polar equation for a conic section, specifically a parabola. The standard form of a parabola in polar coordinates is given by \ r = \frac{k}{1 - e\cos\theta} with \(e = 1\).

Step 2: Recognize parameter values

here, compare the given equation \( r = \frac{1}{1 - \cos\theta} \) with the standard form. We can see that \(e=1\), indicating it's a parabola, and k is also 1.

Step 3: Plot Points

To graph a polar equation, determine coordinates by substituting values of \(\theta\) into the equation. Use values covering important points between 0 and 2\(\pi\) e.g., 0, \(\pi/2\), \(\pi\), 3\(\pi/2\), and 2\(\pi\).

Step 4: Evaluate and plot specific points

Evaluate points:\(\theta=0\), \(r = \frac{1}{1-\cos 0}=\frac{1}{0} \rightarrow \infty\)\ \(\theta = \pi/2\), \(r = \frac{1}{1-0}=1 \)\(\theta=\pi\), \(r = \frac{1}{1-(-1)}=\frac{1}{2}\)\(\theta=3\pi/2\), \(r = \frac{1}{1-0}=1\)

Step 5: Draw the Graph

Using the points calculated in Step 4, plot them in polar coordinates and sketch the parabola. Remember that for \( \theta = 0\) , \( r \) tends to infinity, indicating the graph extends indefinitely in that direction.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. Based on the angle and position of the intersection, four types of conic sections can be produced: circles, ellipses, parabolas, and hyperbolas. Each of these has unique properties and equations.

The Parabolas Concept

A parabola is a specific type of conic section defined as the set of all points that are equidistant from a fixed point called the focus and a line called the directrix. In polar coordinates, a parabola is characterized by its eccentricity (\(e\)) equal to 1. The general equation of a parabola in polar form is given by: \[ r = \frac{k}{1 - \, e \cos \theta} \]. Here, k is a constant, and for parabolas, e = 1. Therefore, the equation simplifies to \[ r = \frac{k}{1 - \cos\theta} \]. By comparing the given equation \( r = \frac{1}{1 - \, \cos \theta} \), we see that it represents a parabola with k = 1.

Graphing Polar Coordinates

Graphing polar coordinates involves plotting points based on their distance from the origin (radius, \(r\)) and the angle from the positive x-axis (\(\theta\)). To graph the polar equation \( r = \frac{1}{1 - \, \cos \theta} \), follow these steps:

• Substitute different values of \(\theta\) (e.g., 0, \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), 2\(\pi\)) to find corresponding \(r\) values.
• For \(\theta = 0\), \( r = \frac{1}{1 - \, \cos 0} \) results in \(r\) being infinite, indicating the graph extends indefinitely in that direction.
• For \(\theta = \frac{\pi}{2}\), \( r = \frac{1}{1 - \, \cos \frac{\pi}{2}} \) equals 1, providing the point (1, \(\frac{\pi}{2}\)).
• For \(\theta = \pi\), \( r = \frac{1}{1 - (-1)} = \frac{1}{2} \), giving the point (0.5, \(\pi\)).
• For \(\theta = \frac{3\pi}{2}\), \( r = \frac{1}{1 - 0} \) equals 1 again.

After calculating these points, plot them on polar coordinates and connect the dots to sketch the parabola.