Problem 8 - Essential Calculus

Question

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{6}{1+\cos \theta}\)

Step-by-Step Solution

Answer

The eccentricity of the conic section is 1 and the distance from the pole to the directrix is 6. This implies that the conic section is a parabola. The graph of the function also confirms this.

Step 1: Match the given polar equation to the general form

We can compare the given equation \(r=\frac{6}{1+\cos \theta}\) with the standard forms \(r=\frac{p}{1+ e \cos \theta}\). Here, we can see, \(p=6\) which stands for the distance from the pole to the directrix and \(e=1\) standing for the eccentricity.

Step 2: Find the eccentricity and distance from the pole to the directrix

The eccentricity 'e' of the conic section is 1 and the distance 'p' from the pole to the directrix is 6. This means that the conic section is a parabola because for a parabola, the eccentricity is 1.

Step 3: Sketch the graph

Plotting the function would yield a graph highlighting the parabola, confirming that with its eccentricity equal to 1 and the distance from the pole to the directrix to be 6, the function does indeed form a parabola. A graphing utility can be used to sketch this graph.

Step 4: Using a graphing utility to confirm

We should then input the equation \(r=\frac{6}{1+\cos \theta}\) into the graphing utility and then observe that the resulting graph is a parabola. This confirms the results obtained in the previous steps.

Key Concepts

Eccentricity of Conic Sections

The term 'eccentricity' is a measure of how much a conic section deviates from being circular. In mathematical terms, it is a non-negative real number that uniquely characterizes a conic section. Conic sections, which arise from the intersection of a plane with a cone, include ellipses, circles, parabolas, and hyperbolas.

For circles, the eccentricity is exactly 0, indicating a perfect circular shape. Ellipses have an eccentricity greater than 0 but less than 1, with values getting closer to 1 as the shape becomes more elongated. In your exercise, the given polar equation corresponds to a conic section with an eccentricity of 1, which unmistakably identifies it as a parabola. Parabolas are unique among conic sections as they have an eccentricity exactly equal to 1. Lastly, hyperbolas have an eccentricity greater than 1, reflecting their open, divergent arms.

Understanding the eccentricity helps in visualization and categorization of conic sections, allowing for a deeper comprehension of their geometrical properties. When given a polar equation, identifying the eccentricity is essential in sketching the graph and determining the shape of the curve.

Distance from Pole to Directrix

When dealing with conic sections in a polar coordinate system, the concept of a directrix plays a pivotal role. The directrix of a conic section is a fixed reference line that, along with a point called the focus, defines the shape of the conic. The distance from the pole (origin of the polar coordinate system) to the directrix, often denoted as 'p', is particularly important for graphing the conic section.

In the equation from the exercise, represented as \(r = \frac{6}{1 + \cos \theta}\), the value of 'p' is 6. This number indicates how far the directrix is from the pole. For a parabola, which we have established is the case due to an eccentricity of 1, this distance determines the 'latus rectum', the width of the parabola. Interestingly, for parabolas, the focus is at the pole, and the directrix is perpendicular to the axis of the parabola and is 'p' units away from the focus.

By examining the distance to the directrix, we can gain insight into the overall size and orientation of the conic section on the polar graph, which is an indispensable step in sketching and understanding the graph's structure.

Sketching Polar Graphs

Sketching polar graphs requires an understanding of the polar coordinate system, where each point on the plane is determined by a distance and an angle from a fixed point known as the pole. To sketch the graph of a conic section, like the parabola from the exercise, first determine the shape of the curve through the eccentricity. Next, understand the role of the directrix and its distance from the pole to visualize the conic's dimensions.

The provided equation \(r = \frac{6}{1 + \cos \theta}\) yields a parabola upon plotting. You'll want to plot points for various angles \(\theta\) and calculate the corresponding 'r' to identify the curve's path. Remember that because the eccentricity is 1, the curve will mirror around the axis of the parabola, which is the line through the focus and perpendicular to the directrix. The distance from the pole to the directrix tells you how wide the parabola opens.

When graphing by hand, start at commonly known angles like 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\), and work your way around the graph, calculating and plotting additional points for a more accurate shape. Modern technology also allows for the use of graphing utilities to check your work, confirming the structure and correctness of the manually sketched graph.