Question

Graphical Reasoning In Exercises 1-4, use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. \(r=\frac{2 e}{1-e \sin \theta}\)

Short answer

For \(e=1\), the graph represents a conic section known as a parabola. For \(e=0.5\), the graph will represent a conic section known as an ellipse. For \(e=1.5\), the graph represents a conic section known as a hyperbola.

Step-by-step solution

Graphing the equation for \(e = 1\)

Substitute \(e = 1\) into the polar equation to get \(r=\frac{2(1)}{1-\sin \theta} = \frac{2}{1-\sin \theta}\). Then graph this equation using a graphing utility.

Identify the conic for \(e = 1\)

Based on the generated graph, identify the type of the conic section. For \(e=1\), the graph represents a conic section known as a parabola.

Graphing the equation for \(e = 0.5\)

Substitute \(e = 0.5\) into the polar equation to obtain \(r=\frac{2(0.5)}{1-0.5 \sin \theta} = \frac{1}{1-0.5\sin \theta}\). Graph this equation using the graphing utility.

Identify the conic for \(e = 0.5\)

Analyze the graph obtained in the previous step. Here, for \(e=0.5\), the graph will represent a conic section known as an ellipse.

Graphing the equation for \(e = 1.5\)

Substitute \(e = 1.5\) into the polar equation to get \(r=\frac{2(1.5)}{1-1.5 \sin \theta} = \frac{3}{1-1.5\sin \theta}\). Graph this equation with a graphing utility.

Identify the conic for \(e = 1.5\)

Analyse the graph generated in the previous step. For \(e=1.5\), the graph represents a conic section known as a hyperbola.

Key concepts

Conic Sections

When learning about conic sections, it's essential to understand that these shapes are the curves obtained by intersecting a plane with a double-napped cone. These intersections create different types of conic sections, such as circles, ellipses, parabolas, and hyperbolas, each with unique mathematical properties and graphs.

The polar equation given in the exercise, \( r=\frac{2e}{1-e \sin \theta} \), can describe different conic sections based on the value of \(e\), which is the eccentricity. For \(e=1\), the curve is a parabola, a u-shaped curve with a single vertex. When \(e<1\), we have an ellipse, an oval shape with two foci. As \(e\) approaches zero, the ellipse becomes more circle-like. Finally, for values of \(e>1\), the equation generates a hyperbola, which consists of two open arms extending in opposite directions.

To identify conic sections given in polar form, one must consider the eccentricity and how it modifies the denominator of the polar equation. For practical learning, graphing these equations with a graphing utility allows visualization of the different shapes, enhancing comprehension of how \(e\) changes their characteristics.

Graphing Utility

A graphing utility is an indispensable tool for visualizing complex mathematical equations, such as those involving polar coordinates and conic sections. Graphing utilities can be software programs or graphing calculators that allow students to input an equation and see its graphical representation instantly.

To get the most out of a graphing utility, you should familiarize yourself with its features: how to enter equations, adjust viewing windows, and interpret the resulting graphs. For the given exercise, using a graphing utility simplifies the task of sketching conic sections by handling the computations and providing an accurate depiction of each curve based on the value of \(e\).

Important tips for using a graphing utility include checking for any asymptotes, symmetry, and intercepts, as these will provide insight into the behavior of conic sections. This technological aid empowers students to experiment with different values quickly and to observe changes in real-time, solidifying their understanding through visual learning.

Polar Coordinates

The polar coordinate system is another way to represent points in a plane, using a distance and an angle, unlike the Cartesian system which uses x and y coordinates. A polar coordinate \( (r, \theta) \) consists of \(r\), the radial distance from the origin, and \(\theta\), the angle measured from the positive x-axis.

Learning how to plot polar coordinates involves moving outwards from the origin a distance of \(r\) and then rotating counter-clockwise by the angle \(\theta\). This system is particularly useful for dealing with problems with circular or rotational symmetry. Polar equations, such as the one you're working with in the exercise, often lead to more natural expressions for spirals, circles, and other curves than their Cartesian counterparts.

Understanding polar coordinates is critical for graphing polar equations efficiently and accurately. By practicing converting between Cartesian and polar forms and plotting various polar equations, students gain a deeper appreciation for the diverse ways we can describe geometric figures and motion paths.