Problem 30 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the type of conic represented by the polar equation and analyze its graph. Then use a graphing utility to graph the polar equation. $$r=\frac{-4}{-1+\cos \theta}$$

Step-by-Step Solution

Answer

The given polar equation represents a parabola. The graph as per the graphing utility would show a curve typical of a parabola.

Step 1: Interpret the given equation

We have the equation \(r=\frac{-4}{-1+\cos \theta}\). We can simplify this to get \(r =\frac{4}{1-\cos \theta}\). This resembles the standard form of a polar equation for a conic section \(r = \frac{e}{1 \pm e cos \theta}\)

Step 2: Identify the conic section by calculating e

In this case, the eccentricity 'e' will be the coefficient of the denominator (1 - cos(theta)). Thus, e = 1. Since |e| = 1, the conic section is a parabola.

Step 3: Graph the equation using a graphing utility

To plot the graph, we must consider a range of theta values and compute the corresponding 'r' values. Use a graphing utility to generate the polar plot for this equation, which will illustrate its parabolic nature.

Key Concepts

Polar Equations

Polar equations are a way of expressing curves and conics using polar coordinates, which are based on angles and distances from a fixed point, known as the pole (or origin). Unlike Cartesian coordinates that use the x and y axes, polar coordinates use 'r' for the distance from the pole and 'θ' (theta) for the angle from the positive x-axis. Polar equations are particularly useful for graphing circles, spirals, and other shapes centered around a point.

In the context of conic sections, we often see polar equations in the form of:

• \( r = \frac{ke}{1 \pm e \cos \theta} \) for conics with a directrix perpendicular to the polar axis
• \( r = \frac{ke}{1 \pm e \sin \theta} \) when the directrix is parallel to the polar axis

Identifying the type of conic (circle, ellipse, parabola, or hyperbola) depends on the value of the eccentricity 'e' in these equations.

Eccentricity

Eccentricity, denoted by 'e', is a number that characterizes the shape of a conic section. It's a crucial concept in understanding the nature of the conic. Here’s how eccentricity helps identify conics:

• If \( e = 0 \), it represents 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 exercise, after simplifying the polar equation to \( r = \frac{4}{1 - \cos \theta} \), the eccentricity was found to be 1. Therefore, the graph of the polar equation represents a parabola.

Understanding eccentricity is key to distinguishing between these conic sections when analyzing their equations. It gives us a simple way to describe the shape and nature of the conics.

Parabola Graphing

Graphing a parabola can seem tricky at first, especially in polar coordinates, but it becomes manageable with the right tools and understanding. In polar graphs, the parabola opens either horizontally or vertically, depending on the orientation indicated by the equation. In our given exercise, the equation \( r = \frac{4}{1 - \cos \theta} \) implies the parabola opens horizontally.

To graph:

• Choose a range of \( \theta \) values. Commonly, \( \theta \) ranges from 0 to \( 2\pi \) or from 0 to \( \pi \), depending on the symmetry.
• Compute corresponding 'r' values for each \( \theta \) to construct the plot.
• Plot these points using a graphing utility capable of polar coordinates.

As you plot the points, the characteristic "U" shape of the parabola becomes evident. Modern graphing utilities make it simple to visualize how these equations translate into the familiar parabolic curves.