Problem 42 - Precalculus Mathematics for Calculus

Question

(a) Graph the conics $$ r=\frac{e d}{(1+e \sin \theta)} $$ for \(e=1\) and various values of \(d .\) How does the value of \(d\) affect the shape of the conic? (b) Graph these conics for \(d=1\) and various values of \(e\) How does the value of \(e\) affect the shape of the conic?

Step-by-Step Solution

Answer

\(d\) changes the size of the parabola; \(e\) changes the conic type.

Step 1: Understanding the Problem

We have a polar equation of a conic section given by \(r = \frac{e d}{1+e \sin \theta}\). Our task is to graph this equation for different parameters \(e\) and \(d\), and observe how these parameters influence the graph's shape. Specifically, we'll treat \(e\) and \(d\) both as constants in separate parts of the analysis.

Step 2: Analyzing the Effect of Parameter \(d\) with \(e = 1\)

Set \(e = 1\) in the conic equation, which simplifies it to \(r = \frac{d}{1 + \sin \theta}\). This signifies a parabola since \(e = 1\). Vary the constant \(d\) and plot the resulting curves for different \(d\) values. The size of the parabola will change, but its basic shape as a parabola remains the same. Larger values of \(d\) make the parabola larger and extend further from the pole.

Step 3: Graphing for Different \(d\) Values and Interpreting Results

Select various values such as \(d = 1, 2, 3\). Plot the resulting graphs using a tool like Desmos or a graphing calculator. Observe that while all graphs are parabolas (since \(e = 1\)), the size of these parabolas increases with \(d\), indicating that \(d\) impacts the scale or size of the conic without altering its type.

Step 4: Analyzing the Effect of Parameter \(e\) with \(d = 1\)

Set \(d = 1\) and focus on how changes in \(e\) affect the conic's shape. The equation becomes \(r = \frac{e}{1 + e \sin \theta}\). The value of \(e\) determines the type of conic: \(e < 1\) (ellipse), \(e = 1\) (parabola), and \(e > 1\) (hyperbola). Vary \(e\) and observe the change in conic types.

Step 5: Graphing for Different \(e\) Values and Interpreting Results

Graph the conic for different \(e\) values, such as \(e = 0.5, 1, 1.5\). For \(e = 0.5\), the graph is an ellipse. For \(e = 1\), the graph is a parabola. For \(e = 1.5\), the graph is a hyperbola. The eccentricity \(e\) directly impacts the conic type, transitioning from ellipse to parabola to hyperbola as \(e\) increases.

Key Concepts

Polar Coordinates

Polar coordinates provide an alternative method to describe points in a plane using two values: radius and angle. In polar coordinates, a point is defined as \((r, \theta)\), where \(r\) represents the radial distance from the origin (or pole), and \(\theta\) represents the angle measured counterclockwise from the positive x-axis.
Key characteristics:

• The radial distance \(r\) can be positive or negative. A negative \(r\) indicates the point is in the opposite direction of \(\theta\).
• Angle \(\theta\) can be expressed in radians or degrees.
• Polar coordinates are especially convenient for dealing with circular and spiral shapes, making them ideal for graphing conic sections like circles and ellipses.

When graphing conics, converting the standard conic equation into polar form can simplify the plotting process, revealing unique traits depending on the eccentricity and other parameters.

Eccentricity

Eccentricity (\(e\)) is a crucial parameter in determining the shape of a conic section in polar coordinates. It measures the deviation of the conic from being circular. Here's how eccentricity affects different conics:

• Ellipse: When \(e < 1\), the conic is an ellipse. The lower the value of \(e\), the more circular the ellipse becomes.
• Parabola: When \(e = 1\), the conic is a perfect parabola. This is a transitional shape, neither bulged nor pinched.
• Hyperbola: When \(e > 1\), the conic is a hyperbola. Increasing \(e\) creates a more open and elongated shape.

Eccentricity helps in classifying conics and predicting their shapes. By adjusting \(e\), one can transition smoothly from one type of conic section to another. In practical applications, understanding \(e\) allows for the prediction and design of paths, such as planetary orbits which are generally elliptical (\(e < 1\)).

Graphing Conics

Graphing conics involves plotting curves based on their polar equation, tailored by the values of eccentricity \(e\), and the semi-latus rectum \(d\). Here's a simple process to graph conics using polar coordinates:- **Identify key parameters**: Retrieve \(e\) and \(d\) from the conic's polar equation, recognizing how they dictate the shape.- **Determine the conic type**: Use the value of \(e\) to identify whether the plot will be an ellipse, parabola, or hyperbola.- **Plot for variation**: Adjust \(d\) or \(e\) and observe changes in the graph.When considering \(e = 1\) and varying \(d\), you produce a series of parabolas that expand as \(d\) increases. Conversely, keeping \(d = 1\) and varying \(e\) shifts the conic type:

• from ellipse \((e < 1)\)
• to parabola \((e = 1)\)
• to hyperbola \((e > 1)\)

Thus, playing with these parameters is key to understanding and visualizing different conic sections.