Problem 29 - 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. $$r=\frac{-3}{-4+2 \cos \theta}$$

Step-by-Step Solution

Answer

The given polar equation represents an ellipse. The graph would be a complete ellipse having the origin as one of the foci, horizontal orientation, and eccentricity of \( \frac{3}{2}\).

Step 1: Identifying the conic section

In polar coordinates, the conic sections are usually represented as \(r = \frac{ed}{1 \pm e \cos(\theta - \phi)}\), where e is the eccentricity, d is the distance from the origin to the directrix, \(\theta\) is the angle and \(\phi\) is provided angle. Compare the given equation \(r=\frac{-3}{-4+2 \cos \theta}\) to the standard forms. Notice it can be rewritten as \(r = \frac{3}{2(2 - \cos \theta)}\). Here, \( e = \frac{3}{2}\), presenting an ellipse since the e is less than 1.

Step 2: Plotting the polar function

Now, you will graph the equation \(r = \frac{3}{2(2 - \cos \theta)}\) using a graphing utility. This step depends on the specific graphing tool being used, but in most you will input the equation exactly as given. Be sure to set the mode to 'polar' as the equation is in polar coordinates rather than Cartesian.

Step 3: Analyzing the graph

Finally, you analyze the graph. In general, you should look for key features of the plot that correspond to the properties of the ellipse. This would include noting the shape of the plot, looking for symmetry, noting the length and width of the ellipse, as well as the orientation in the polar plane.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a right circular cone with a plane. Traditionally, these curves include ellipses, parabolas, hyperbolas, and circles. A unique property of these shapes is that they can be represented in both Cartesian and polar coordinates.

In polar coordinates, the general form of a conic section is given by the equation:

• \( r = \frac{ed}{1 \, \pm \, e \cos(\theta \, - \, \phi)} \)

Here, \(e\) represents the eccentricity, \(d\) is the distance from the focus to the directrix, and \(\theta - \phi\) is the angle.

The type of the conic is determined by the eccentricity \(e\). Specifically, if

• \(e = 0\), the conic is a circle.
• \(0 < e < 1\), it forms an ellipse.
• \(e = 1\), it's a parabola.
• \(e > 1\), you're dealing with a hyperbola.

Eccentricity

Eccentricity is a number that helps to define the shape of a conic section. It denotes how much the conic section deviates from being circular. When working with polar equations, calculating \(e\) is crucial for classifying the conic.

To find eccentricity from a polar equation, compare the given equation to the standard form of conic in polar coordinates. Once you have the standard form, determine \(e\) by looking at the coefficient of \(\cos\theta\) in the denominator.

• For an ellipse, as in this exercise, \(e\) is between 0 and 1, showing a lesser deviation from circularity than other conics.
• The closer \(e\) is to 0, the more circular the ellipse appears.

In the given example, we found \(e = \frac{3}{2}\). This value, less than 1, confirms that the conic is indeed an ellipse.

Ellipse

An ellipse is a type of conic section that looks like a stretched circle. In simpler terms, it is an "oval" shape. In polar coordinates, an ellipse has certain known properties.

The most important one is that the sum of the distances from any point on the ellipse to the two foci is constant. This characteristic helps to define the shape precisely.

In the equation \(r = \frac{ed}{1 \, - \, e \cos(\theta)}\), the eccentricity \(e < 1\) confirms the shape is an ellipse. If you rewrite the exercise's equation in this form, it clearly indicates an ellipse, given that \(e = \frac{3}{2}\), below one. This defines its more circular nature compared to other conic sections.

Graphing Polar Equations

Graphing polar equations can initially seem tricky, but it's easier with some practice. Polar coordinates express points in the form \(r, \theta\), with \(r\) being the radius and \(\theta\) being the angle.

To graph a polar equation such as \(r = \frac{3}{2(2\, - \, \cos \theta)}\), we utilize a graphing utility or calculator.

Here's how to proceed:

• Set the calculator to 'polar' mode.
• Input the polar equation directly.
• Adjust the settings to capture the full range of \(\theta\), typically from \(0\) to \(2\pi\).

After graphing, analyze the output to observe symmetry and the elliptical nature of the plot. Observations such as the major and minor axes help in understanding the size and orientation of the ellipse within the polar plane.