Problem 73 - Algebra and Trigonometry

Question

Graph each polar equation. $$ r=\frac{2}{1-\cos \theta} \quad(\text {parabola}) $$

Step-by-Step Solution

Answer

Plot points for specific angles and sketch the parabola in the polar coordinate system.

Step 1 - Identify the Polar Equation Form

The given polar equation is in the form the form: \[ r = \frac{e \times k}{1 - e \cos \theta} \]which represents a conic section with focus at the pole

Step 2 - Recognize the Conic Section

Identify the type of conic. For the given equation, we have: \( r = \frac{2}{1 - \cos \theta} \). This matches the standard form of a parabola because the denominator is a linear function of \( \cos \theta \).

Step 3 - Determine Parameters

For \( r = \frac{2}{1 - \cos \theta} \), we can see that the eccentricity \( e = 1 \) which confirms its a parabola. The directrix is a vertical line located at \( x = 2 \).

Step 4 - Plot Key Points

Determine points for several values of \( \theta \): \( \theta = 0 \), \( r = \frac{2}{1 - \cos 0} \) becomes \( \frac{2}{1 - 1} \rightarrow \text{undefined} \). \( \theta = \frac{\pi}{3} \), \( r = \frac{2}{1 - \cos \frac{\pi}{3}} = 2 \). \( \theta = \frac{\pi}{2} \), \( r = \frac{2}{1 - \cos \frac{\pi}{2}} = 2 \). \( \theta = \pi \), \( r = \frac{2}{1 - \cos \pi} = 1 \). These points can be plotted with their respective coordinates.

Step 5 - Sketch the Polar Graph

Place the points on polar coordinate graph paper. Since it's a parabola, connect the points accordingly to show an open-ended curved shape. Verifying that the curve gets close to the directrix line at \( x = 2 \) without touching it.

Key Concepts

Understanding Polar Equations

Polar equations describe curves using the polar coordinate system, where each point is determined by a distance from the origin and an angle from a fixed direction. This system is different from Cartesian coordinates, which use x and y coordinates. The polar form of a conic section is generally given by:
\[ r = \frac{e \times k}{1 - e \, \cos \, \theta} \] Here, **r** is the radius, **θ** is the angle, **e** is the eccentricity, and **k** is a constant. Each value of θ produces a corresponding r, allowing us to plot the curve. The form of the equation can tell us a lot about the nature of the graph, such as its symmetry and type.

Exploring Conic Sections

Conic sections are shapes created by the intersection of a plane and a double-napped cone. These shapes include parabolas, ellipses, and hyperbolas. In our polar equation, the form of the conic section is determined by the value of **e** (the eccentricity).

• If **e = 1**, the conic section is a **parabola**.
• If **e < 1**, it is an **ellipse**.
• If **e > 1**, it is a **hyperbola**.

The pole (or the origin in a Cartesian system) acts as the focus for these conic sections. In our given equation, **e = 1**, confirming that it is a parabola. The line **x = 2** then acts as the directrix, a key reference line for plotting the parabola.

Graphing a Parabola in Polar Coordinates

A parabola's unique feature is that each point on it is equidistant from the focus and the directrix. In this exercise, the parabola is described by:
\[r = \frac{2}{1 - \cos(\theta)}\]
This means with **e = 1**, we are dealing with a parabola. To graph it, follow these steps:

• Calculate r for various θ values. For example:
  • \(θ = 0\) gives \( r = \frac{2}{1 - \cos(0)} \rightarrow \text{undefined}\)
  • \(θ = \frac{\pi}{3}\) gives \( r = \frac{2}{1 - \cos(\frac{\pi}{3})} = 2\)
• Plot these points on polar graph paper, noting the position and distance from the origin.
• Connect the points to form an open-ended curved shape approaching the directrix but never touching it.

By understanding the relationship between θ and r, you can accurately sketch the curve of the parabola in a polar coordinate system.