Problem 69 - Thomas Calculus

Question

Use a CAS to plot the polar equation $$ r=\frac{k e}{1+e \cos \theta} $$ for various values of \(k\) and \(e,-\pi \leq \theta \leq \pi .\) Answer the following questions. a. Take \(k=-2 .\) Describe what happens to the plots as you take \(e\) to be \(3 / 4,1,\) and \(5 / 4 .\) Repeat for \(k=2 .\) b. Take \(k=-1 .\) Describe what happens to the plots as you take \(e\) to be \(7 / 6,5 / 4,4 / 3,3 / 2,2,3,5,10,\) and \(20 .\) Repeat for \(e=1 / 2,1 / 3,1 / 4,1 / 10,\) and \(1 / 20 .\) c. Now keep \(e>0\) fixed and describe what happens as you take \(k\) to be \(-1,-2,-3,-4,\) and \(-5 .\) Be sure to look at graphs for parabolas, ellipses, and hyperbolas.

Step-by-Step Solution

Answer

For varying \( e \), ellipses, parabolas, and hyperbolas emerge; varying \( k \) affects size/orientation. Eccentricity less than 1 gives ellipses, equal to 1 gives parabolas, and greater than 1 gives hyperbolas.

Step 1: Understanding the Polar Equation

The given polar equation is \( r = \frac{k e}{1 + e \cos \theta} \). This represents conic sections as polar graphs, with the eccentricity \( e \) affecting the shape (ellipse, parabola, or hyperbola), and \( k \) affecting the size and orientation.

Step 2: Setup for Plotting

Before plotting, recognize that different values of \( k \) and \( e \) will convert the equation into various conics. For \( e < 1 \), the graph is an ellipse; for \( e = 1 \), it's a parabola; and for \( e > 1 \), it's a hyperbola. The parameter \( k \) affects the size and orientation without changing these characteristics.

Step 3: Plotting for k = -2, e = 3/4, 1, 5/4

Using a CAS, plot the equation for \( k = -2 \) and \( e = \frac{3}{4}, 1, \frac{5}{4} \). For \( e = \frac{3}{4} \), expect an ellipse, which becomes narrower as \( e \) approaches 1 (parabola), and then becomes a hyperbola for \( e = \frac{5}{4} \). Each plot's size is scaled by \( k = -2 \), flipping it across the pole.

Step 4: Plotting for k = 2, e = 3/4, 1, 5/4

For \( k = 2 \) and the same values of \( e \), the graph changes in size and flips because of the positive \( k \). Ellipse at \( e = \frac{3}{4} \), parabola at \( e = 1 \), and hyperbola at \( e = \frac{5}{4} \) are observed, similar in shape to the previous step but flipped.

Step 5: Plotting for k = -1, varying e (7/6 to 20)

Use a CAS to plot for \( k = -1 \) and increasing \( e \): \( \frac{7}{6}, \frac{5}{4}, \frac{4}{3}, \frac{3}{2}, 2, 3, 5, 10, 20 \). For small \( e > 1 \), the hyperbola's branches are closer; increasing \( e \) causes them to widen, moving farther apart.

Step 6: Plotting for k = -1, decreasing e (1/2 to 1/20)

Plot for \( k = -1 \) and decreasing \( e \): \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{10}, \frac{1}{20} \). As \( e \) decreases, the ellipse becomes more circular, with smaller \( e \) making the ellipse nearly a circle.

Step 7: Fixing e, Varying k (-1 to -5)

With fixed \( e > 0 \), vary \( k = -1, -2, -3, -4, -5 \). Observe how the overall size decreases as \( |k| \) increases, affecting the distance of the conic sections from the pole without changing their type.

Key Concepts

Conic Sections

Conic sections are the curves obtained by slicing a double cone and consist of circles, ellipses, parabolas, and hyperbolas. These shapes are everywhere in mathematics and engineering and can also be beautifully expressed as polar equations. When we map these shapes using polar coordinates, our polar equation becomes \( r = \frac{k e}{1 + e \cos \theta} \), where the eccentricity \( e \) helps determine which conic section we will see.

• Ellipses: Occurs when \( e < 1 \). They are closed shapes that resemble elongated circles.
• Parabolas: Occurs when \( e = 1 \). These are open, symmetrical curves, like satellite orbits.
• Hyperbolas: Occurs when \( e > 1 \). These are open with two branches going in opposite directions.

The constant \( k \) modifies the size and position of these conic sections without changing their basic form. As \( k \) changes, we observe changes in orientation and scale of these conic beings, flipping them over the pole when \( k \) is negative.

Eccentricity

The concept of eccentricity measures how much a conic section deviates from being circular. In simpler words, it tells us the "stretchiness" or openness of the conic shape.

• Ellipses have eccentricity \( e < 1 \), showing less deviation from a circle.
• A circle specifically has an eccentricity of 0, an extreme case of an ellipse.
• Parabolas sit at the boundary case with \( e = 1 \), indicating a perfect balance between openness and alignment.
• Hyperbolas, flaunting \( e > 1 \), use higher values of eccentricity to show their dramatic spread.

Changing the eccentricity affects the appearance: a higher \( e \) in hyperbolas widens their branches further apart, while a lower \( e \) in ellipses makes them more circle-like. It is a key factor in defining the type of conic in our polar equation, \( r = \frac{k e}{1 + e \cos \theta} \). Understanding eccentricity means immediately recognizing the type of conic you're dealing with.

Parametric Plotting

Parametric plotting offers a convenient way to visualize curves that can be difficult to express explicitly. In the space of polar equations, parametric plots can offer clarity.
When plotting the equation \( r = \frac{k e}{1 + e \cos \theta} \) for different values of \( k \) and \( e \), we focus on how the parameters affect the shape and position of the plot. Using a Computer Algebra System (CAS) can assist greatly, allowing for interactive exploration of these curves.

• By changing \( k \), see how the size and orientation flips when plotting. Negative \( k \) values flip the graph over, whereas positive values retain the original spread.
• Adjusting \( e \) alters the openness (for hyperbolas) or closed nature (for ellipses) of the plot.
• As we fix one parameter and vary the other, CAS tools can show continuous transformations between these shapes, providing visual answers to theoretical questions.

Engaging with parametric plotting not only supplements algebraic understanding but also builds intuition for the space and structure of conic sections.