Question

Writing Consider the polar equation \(r=\frac{4}{1+e \sin \theta} .\) (a) Use a graphing utility to graph the equation for \(e=0.1\), \(e=0.25, e=0.5, e=0.75,\) and \(e=0.9 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{-}\) and \(e \rightarrow 0^{+}\) (b) Use a graphing utility to graph the equation for \(e=1\). Identify the conic. (c) Use a graphing utility to graph the equation for \(e=1.1\), \(e=1.5,\) and \(e=2 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{+}\) and \(e \rightarrow \infty\).

Short answer

The short answer to this question can be understood by plotting the equation for different values of \( e \) as described. For \( e < 1 \), the conic section appears to be an ellipse with increasing eccentricity as \( e \) gets closer to 1. For \( e = 1 \), the graph is a parabola. When \( e > 1 \), the graph devolves into a hyperbola, getting increasingly 'open' as \( e \) increases toward infinity.

Step-by-step solution

Graphing for \( e < 1 \)

Use the graphing tool to draw the equation \( r=\frac{4}{1+e \sin \theta} \) for each of the following values of \( e \): 0.1, 0.25, 0.5, 0.75, and 0.9. Observe how the shape of the conic section changes as \( e \) approaches 1 from the left and 0 from the right.

Graphing for \( e = 1 \)

Graph the equation \( r=\frac{4}{1+e \sin \theta} \) for \( e = 1 \). Identify the conic section.

Graphing for \( e > 1 \)

Lastly, graph the equation \( r=\frac{4}{1+e \sin \theta} \) for \( e \) values of 1.1, 1.5, and 2. Like before, identify the conic section and notice the changes in the shape as \( e \) approaches 1 from the right and goes to infinity.

Key concepts

Conic Sections

When we talk about conic sections in mathematics, we are referring to the curves obtained as cross sections of a cone. These include circles, ellipses, parabolas, and hyperbolas, each distinguished by their unique shape and equations. A vital concept here is the eccentricity (\( e \)), which determines the specific type of conic section represented based on its value.

The polar equation stated in the exercise, \( r = \frac{4}{1 + e \sin(\theta)} \), describes a conic section with a focus at the origin and directrix perpendicular to the polar axis. As the exercise suggests, varying the eccentricity will alter the conic section's shape:

• For \( 0 \le e < 1 \), the graph will represent an ellipse, including a circle when \( e = 0 \).
• When \( e = 1 \), the conic section is a parabola.
• For \( e > 1 \), it results in a hyperbola.

The changes in the shape of the graph become more pronounced as \( e \) approaches the defining limits of eccentricity, transforming how each conic section is presented.

Graphing Utilities

Graphing utilities are immensely valuable tools in visualizing mathematical concepts, especially conic sections. They convert complex equations into visual graphs, making the understanding of shapes and behaviors much more intuitive. In the exercise, graphing utilities are used to plot the equation \( r = \frac{4}{1 + e \sin(\theta)} \) for various values of eccentricity.

For students, graphing these polar equations using software or a graphing calculator allows them to experiment and see real-time changes in the graph as they adjust the value of \( e \). It is a practical way to grasp the relationship between the polar equation and its corresponding conic section. For educators, this hands-on approach can be an excellent teaching aid to demonstrate the transition between different types of conics—a circle morphing into an ellipse and eventually turning into a parabola or hyperbola, as the eccentricity increases or decreases.

Limits of Eccentricity

Eccentricity, denoted as \( e \), is a parameter that determines the nature of a conic section. By exploring the limits of eccentricity, we understand the bounds within which these conic sections change. In the polar equation provided, \( r = \frac{4}{1 + e \sin(\theta)} \), as \( e \) tends toward certain values, we witness different conic sections:

• As \( e \rightarrow 0^{+} \), we're approaching the limit where the conic section becomes more circular.
• When approaching \( e \rightarrow 1^{-} \), the ellipse elongates and trends towards becoming a parabola.
• Moving past \( e = 1 \), towards \( e \rightarrow 1^{+} \), the conic transitions from a parabola to a hyperbola.
• As \( e \) increases towards infinity, the arms of the hyperbola become more open and less curved.

The concept of limits in eccentricity helps students predict the behavior of conic sections and understand the continuum of shapes they can take. It's important for students to internalize that the value of \( e \) is not arbitrary; it has a direct, tangible effect on the geometric properties of the curve represented by the given polar equation.