Question

Use a graphing utility to graph the hyperbolas \(r=\frac{e}{1+e \cos \theta},\) for \(e=1.1,1.3,1.5,1.7\) and 2 on the same set of axes. Explain how the shapes of the curves vary as \(e\) changes.

Short answer

Answer: As \(e\) increases, the hyperbolas become more stretched along the horizontal axis, resulting in a larger horizontal distance between the vertices. This also implies that the focal distance of the hyperbolas increases.

Step-by-step solution

Understand the given equation

The given equation is a polar equation, and it represents the equation of a hyperbola where \(r\) is the radial distance and \(\theta\) is the angle. The variable \(e\) is a parameter that influences the shape of the curves.

Graph the hyperbolas

Using a graphing utility (such as Desmos or a graphing calculator), graph the equation \(r=\frac{e}{1+e \cos \theta}\) for each of the given values of \(e\): \(1.1\), \(1.3\), \(1.5\), \(1.7\), and \(2\). Ensure that all hyperbolas are plotted on the same set of axes.

Observe the changes in the shape of the curves

When the graphs are plotted, you will notice that as \(e\) increases, the hyperbolas become more "stretched" along the horizontal axis. This implies that the focal distance (the distance between the foci) of the hyperbolas is also increasing.

Explain the variation in the shapes

The changes in the shape of the curves can be attributed to the parameter \(e\). As \(e\) increases, the equation \(\frac{e}{1+e \cos \theta}\) becomes larger for the same value of \(\theta\), which results in a larger horizontal distance between the vertices of the hyperbola. This causes the hyperbola to be stretched along the horizontal axis. As a conclusion, the shapes of the curves vary as \(e\) changes, becoming more stretched along the horizontal axis as \(e\) increases.

Key concepts

Polar Equations

Polar equations are mathematical representations that define curves using a coordinate system where points are determined by a distance from a central point, known as the pole, and an angle from a fixed direction. Unlike the Cartesian system, which uses x and y coordinates, polar coordinates are expressed as \( (r, \theta) \) where \( r \) is the radius or radial distance from the pole, and \( \theta \) is the angle measured in radians from the polar axis, typically the positive x-axis.

For example, the polar equation for a circle with a radius of \( a \) located at the pole is written as \( r = a \) irrespective of \( \theta \). But with the hyperbolas we are considering in this exercise, the equation takes a different form. Hyperbolas are expressed using parameters such as eccentricity \( e \), leading to the equation \( r = \frac{e}{1 + e \cos \theta} \) where \( e \) determines the shape of the hyperbola. Understanding how to interpret and graph these equations is essential for visualizing complex shapes in polar coordinates.

Conic Sections in Polar Coordinates

Conic sections—a set of shapes including circles, ellipses, parabolas, and hyperbolas—can be elegantly described using polar coordinates. Each conic section's polar equation varies depending on the shape and its position relative to the pole.

When we talk about a conic section like a hyperbola in polar coordinates, it's often with respect to its eccentricity \( e \), which is manifested in the polar equation \( r = \frac{e}{1 + e \cos \theta} \). When \( e > 1 \), the equation represents a hyperbola. For an ellipse, \( e \) would be less than 1, and if \( e \) is exactly 1, the equation would describe a parabola. Hence, mastering polar equations of conic sections is crucial for students to understand the geometric properties related to polar coordinates and how to convert them into Cartesian coordinates if needed.

Graphing Utility Usage

Graphing utilities such as Desmos, GeoGebra, and various graphing calculators are indispensable tools in visualizing mathematical equations, especially for complex curves like hyperbolas in polar coordinates. These utilities can handle the polar equations directly and provide an immediate graphical representation of the curve without manual plotting.

To graph a hyperbola using these tools, simply input the general polar equation and adjust the parameters such as \( e \) as needed. The utility takes care of the rest, mapping out the hyperbola for every angle \( \theta \) and creating a visual that is far easier to comprehend than numbers or algebra alone. By learning efficient usage of these graphing utilities, students can quickly experiment with various equations, observing the impact of changes in parameters on the shape of their graphs in real-time.

Effects of Eccentricity on Hyperbolas

In the case of hyperbolas, eccentricity \( e \) is a crucial parameter indicating how 'stretched' the curve is. For the equation \( r = \frac{e}{1 + e \cos \theta} \), as \( e \) increases, the shape of the hyperbola changes significantly. An \( e \) greater than 1 indicates a hyperbola, where the larger the value of \( e \) is, the more the curve stretches away from the central point.

The given problem illustrates this perfectly. By graphing hyperbolas for \( e \) values of 1.1, 1.3, 1.5, 1.7, and 2, students will observe that the hyperbolas become more elongated as \( e \) grows. This stretching is because for any given angle \( \theta \) the ratio \( \frac{e}{1 + e \cos \theta} \) becomes larger, pushing the vertices of the hyperbola further apart, thus giving it a more 'open' appearance. Understanding the effect of eccentricity on the shape of hyperbolas helps students grasp the broader concept of how conic sections behave under varying parameters.