Problem 20 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the type of conic represented by the equation. Use a graphing utility to confirm your result. $$r=\frac{9}{3-2 \cos \theta}$$

Step-by-Step Solution

Answer

The given polar equation is an ellipse, confirmed by the graph of the equation.

Step 1: Working with the Given Equation

Rewrite the given equation \(r = \frac{9}{3 - 2\cos\theta}\) into a form \(r = \frac{p}{1 - e\cos(\theta)}\). Here \(p = \frac{9}{2}\) and \(e = \frac{2}{3}\).

Step 2: Identifying the Conic Section

After performing the first step, you'll understand that polar equations of conic sections have eccentricity value \(e\) which determines the type of conic. If \(e < 1\), it's an ellipse; if \(e = 1\), it's a parabola; and if \(e > 1\), it's a hyperbola. So, since \(e = \frac{2}{3} < 1\), it is an ellipse.

Step 3: Confirming with a Graph

To confirm the above conclusion, input the equation \(r = \frac{9}{3 - 2\cos\theta}\) into a graphing utility such as GeoGebra. The plotted graph will be an ellipse, which verifies the result.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to describe points in the plane. Instead of using \(x, y\) coordinates like in Cartesian systems, polar coordinates consist of a radius \(r\) and an angle \(\theta\). \(r\) represents the distance from the origin, and \(\theta\) is the angle measured from the positive x-axis.For example, the equation \(r = \frac{9}{3 - 2\cos\theta}\) is expressed in polar form. Here, we manipulate the values of \(r\) depending on the angle \(\theta\). By rewriting this equation into its standard conic section form, we gain insight into the geometric shape it represents.

Ellipse

An ellipse is a type of conic section that you can identify by its eccentricity. It appears as an oval-shaped curve and looks like a stretched circle. In polar coordinate equations, an ellipse can be recognized when the eccentricity \(e\) is less than 1.For instance, in the equation \(r = \frac{9}{3 - 2\cos\theta}\), when it is rewritten in the format \(r = \frac{p}{1 - e\cos\theta}\), you find that \(e = \frac{2}{3}\). Since \(\frac{2}{3} < 1\), this indicates that the graph in question indeed describes an ellipse. The ellipse will be centered at a point known as the directrix in polar form.

Graphing Utility

A graphing utility is a useful tool for visualizing equations. It helps you confirm theoretical predictions by plotting complex equations. Many students and professionals use software such as GeoGebra or Desmos to analyze the behavior of functions and geometric shapes.In the case of the equation \(r = \frac{9}{3 - 2\cos\theta}\), plotting it in a graphing utility shows whether the curve deviates as predicted. By visualizing this equation, you'll observe the shape forms an ellipse, confirming what we deduced analytically. This step is crucial as it gives a visual confirmation that reinforces your understanding.

Eccentricity

Eccentricity is a key concept when dealing with conic sections. It determines the shape and type of the conic section you are dealing with.

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

In our problem, we calculated an eccentricity of \(\frac{2}{3}\), indicating the equation represents an ellipse. Understanding eccentricity is essential because it provides a straightforward way to classify and understand different conic sections, which appear frequently in both nature and mathematical applications.