Question

Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. $$r=\frac{1}{2-\cos \theta}$$

Short answer

Answer: The conic section represented by the polar equation $$r = \frac{1}{2 - \cos \theta}$$ is a parabola. The focus of the parabola is at (1, 0) in polar coordinates, and the directrix is a vertical line with the equation $$x = -1.$$

Step-by-step solution

Identify the conic section

To identify the conic section, compare the given polar equation $$r = \frac{1}{2 - \cos \theta}$$ with the standard equations for each conic section. We notice that the equation matches the standard equation for a conic section in the form $$r = \frac{ep}{1 \pm e\cos \theta}$$ where e is the eccentricity (an ellipse when e < 1, a parabola when e = 1, and a hyperbola when e > 1). Using this, we can find the eccentricity e and the distance p of the directrix from the pole: $$e = 1, \hspace{1cm} p = 1.$$ Since the eccentricity e = 1, the conic section is a parabola.

Find the required properties

Now, we will find the properties of the parabola. Since we know the distance p of the directrix from the pole, we can find the focus and the directrix equation. Focus: The focus will be at the point (p, 0) in polar coordinates, which is \((1, 0)\). Directrix: The directrix will be a vertical line p units to the left of the pole, with the equation being $$x = -1.$$ Since it is a parabola, there are no asymptotes.

Graph the conic section and check with a graphing utility

To graph the polar conic section $$r = \frac{1}{2 - \cos \theta},$$ plot points for various angles θ from 0 to 2π, taking care to include the endpoints and points of interest (e.g., when θ = 0, π/2, π, 3π/2). For example, for θ = 0: $$r = \frac{1}{2 - \cos(0)} = \frac{1}{2 - 1} = 1.$$ Therefore, the point for θ = 0 is \((1, 0)\), which is exactly equal to the focus. Continue to plot points for different angles and connect them to form a smooth curve. Using a graphing utility, verify that the graph is accurate. In conclusion, the graph of the conic section with polar equation $$r = \frac{1}{2 - \cos \theta}$$ is a parabola with focus at \((1, 0)\) and directrix $$x= -1.$$

Key concepts

Parabola

A parabola is a unique curved shape that looks like an open arch. It's one of the most common conic sections you encounter in math. Its defining feature is its eccentricity, denoted as \(e\). For parabolas, this value is exactly 1. This is in contrast to ellipses and hyperbolas which have different eccentricities. The parabola is perfectly symmetrical about its axis, which is a line of symmetry running through the shape.

The parabola has some key components:

• Vertex: This is the point that acts as a peak or a turn in the parabola, often at the lowest or highest point depending on its orientation.
• Focus: This is a special point located inside the parabola. In polar coordinates, the focus of our parabola lies at \((1, 0)\).
• Directrix: This is an imaginary line that is the same distance from any point on the parabola as the focus. For our parabola, that line is \(x = -1\).
• Axis: A vertical or horizontal line that divides the parabola into two mirror-image halves.

Parabolas appear naturally in physics and engineering, like in the path of a thrown object or the design of satellite dishes. Understanding their properties and how to graph them is fundamental in both mathematical theory and practical application.

Polar Coordinates

Polar coordinates offer an alternative way to describe a point in the plane using a radius and an angle, rather than using the traditional x and y coordinates. This system is extremely useful for graphing equations that are naturally circular or spiral, like certain conic sections.

In polar coordinates, a point is defined as \((r, \theta)\):

• \(r\) is the radius, the distance from the origin (known as the pole) to the point.
• \(\theta\) is the angle, measured in radians, from the positive x-axis.

Some important things to consider about polar coordinates are:

• You can have multiple representations for a single point, because angles can be given in multiple rotations (e.g., \(2\pi\) and \(0\) are equivalent).
• The distance \(r\) can be negative, which effectively reflects the point across the origin.

In the context of our exercise, the polar equation \(r=\frac{1}{2-\cos \theta}\) helps us understand how the parabola is set up in a polar coordinate system, emphasizing its relationship to angles and distances from its focus.

Graphing Utilities

Graphing utilities are modern tools that help visualize mathematical concepts and verify the correctness of hand-drawn graphs. These can be either software or calculators designed to graph equations easily and accurately. Using these tools can significantly improve your understanding of complex graphs, like conic sections, by providing a quick, visual representation.

Some popular graphing utilities include:

• Graphing Calculators: Devices like the TI-84 can plot points from given equations and find intercepts, maxima, minima, etc.
• Computer Software: Programs such as Desmos or GeoGebra allow you to input equations and immediately see their graphs. They often have interactive features to manipulate and study geometrical properties.
• Online Tools: Many websites provide simple input boxes where you can enter equations to get instant graphs.

By inputting the polar equation \(r = \frac{1}{2 - \cos \theta}\) into these tools, you can easily verify the plotted parabola with its correct orientation and intersection points. This convenience makes graphing utilities invaluable in learning and validating mathematical concepts.