Question

Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. $$r=\frac{4}{1+\cos \theta}$$

Short answer

Answer: The given polar equation represents a parabola with vertex at (2, 0), focus at (4, 0), and directrix at r=0.

Step-by-step solution

Identify the type of conic section

To identify the type of conic section represented by the given polar equation, we need to determine the eccentricity (\(e\)) of the conic section, which can be found by comparing the given polar equation to the standard eccentricity form for a polar equation: $$r = \frac{p}{1 + e\cos\theta}$$. So, we can see that $$p=4$$ and $$e=1$$. A conic section with an eccentricity of 1 represents a parabola. Step 2: Find the necessary elements of the conic section

Determine the associated elements

Since we are dealing with a parabola, we need to find its vertex, focus, and directrix. We can find these elements using the following formulas: - Vertex: The vertex of a parabola in polar form is at \((r_0, 0)\), where \(r_0 = \frac{p}{2}\). In our case, \(r_0 = \frac{4}{2} = 2\). - Focus: The focus of a parabola is located at \((r_0 + \frac{p}{2}, 0)\). So in our case, it is at \((2+\frac{4}{2}, 0) = (4,0)\). - Directrix: The directrix is a vertical line at \(r=r_0-\frac{p}{2}\). So, its equation is \(r=2-\frac{4}{2}=0\). Step 3: Graph the conic section and label the associated elements

Graph the conic section

To graph the conic section: 1. Plot the vertex, focus, and directrix on a polar coordinate plane. 2. Sketch the parabolic shape, making sure the parabola intersects the vertex and focus and is symmetrical about the directrix. 3. Label the vertex (\(V\)), focus (\(F\)), and directrix. Step 4: Check the graph using a graphing utility

Verify the graph

Use a graphing utility (such as a graphing calculator or a website like Desmos) to plot the given polar equation: $$r=\frac{4}{1+\cos \theta}$$. Ensure that the plotted graph matches the one we sketched in Step 3. Additionally, confirm that the vertex, focus, and directrix are correct.

Key concepts

Polar Coordinates

Polar coordinates offer a unique way to represent points in the plane. Instead of the standard Cartesian coordinates \((x, y)\), polar coordinates use \((r, \theta)\), where \(r\) is the distance from the origin (or pole) and \(\theta\) is the angle from the positive x-axis.
This system is especially useful for graphing conic sections, like circles, ellipses, or parabolas, because it often simplifies the equations. For example, a parabola's polar equation like the one in the exercise \( r = \frac{4}{1+\cos \theta} \) clearly shows how \(r\) depends directly on \(\theta\).

• For \(\theta = 0\), \(r\) is purely determined by the constant \(p=4\) in the numerator
• The denominator's structure \(1 + \cos \theta\) indicates dependence on the orientation of the parabola related to the polar axis.

This makes polar coordinates an effective tool in identifying and analyzing the shapes and orientations of various conic sections.

Parabola

A parabola is a specific type of conic section. It can be recognized in polar form by examining its equation, which resembles \(r = \frac{p}{1 + e\cos\theta}\), where the eccentricity \(e = 1\). In this scenario, the equation indicates that we are dealing with a parabola rather than an ellipse or hyperbola.
Parabolas have unique features:

• Vertex: The starting point of the parabola's curve.
• Focus: A point from which distances to any point on the parabola are measured.
• Directrix: A line that, along with the focus, helps define the parabola's shape and symmetry.

In the exercise, the vertex is at \( (2, 0) \), the focus at \( (4, 0) \), and the directrix aligns with the line \(r = 0\), showing the symmetrical nature of parabolas when plotted in polar coordinates.

Graphing Utility

A graphing utility, such as a graphing calculator or an online tool like Desmos, plays a vital role in visualizing complex equations and their corresponding graphs. These tools help confirm the accuracy of hand-drawn graphs, offering an opportunity to identify errors or further understand the conic section's properties.
To use a graphing utility efficiently:

• Input the accurate equation, ensuring the polar mode is selected for conic sections like parabola.
• Check if plotted points, like the vertex and focus, match manual calculations.
• Observe the general shape and symmetry to ensure it represents the described conic section.

Graphing utilities attend to details, like line curvatures and exact intersections, with precision that might be challenging in manual plotting. They're indispensable for students needing visual confirmation of mathematical predictions.

Eccentricity

Eccentricity is an essential concept in understanding conic sections, particularly parabolas. Eccentricity, denoted by \(e\), is a numerical representation of how much a conic section diverges from being a circle. Here it's calculated within the equation \(r = \frac{p}{1 + e\cos\theta}\). For parabolas specifically, \(e = 1\), a defining feature distinguishing them from ellipses (\(e < 1\)) and hyperbolas (\(e > 1\)).
In this exercise, recognizing \(e=1\) tells us we're working with a parabola.
By knowing \(e\), one can anticipate the shape and behavior of the conic section without plotting it:

• Parabolas appear more elongated, with a symmetrical curve that extends infinitely.
• The location of the vertex, focus, and directrix becomes predictable based on the eccentricity.

Understanding eccentricity introduces a systematic way to categorize and subsequently study conic sections, thus making them comprehensible and predictable.