Problem 24 - Precalculus: Mathematical for Calculus

Question

Graph the parabola \(r=5 /(2+2 \sin \theta)\) and its directrix. Also graph the curve obtained by rotating this parabola about its focus through an angle \(\pi / 6\)

Step-by-Step Solution

Answer

Just graph the parabola and its directrix; rotate it \( \pi/6 \) about the focus.

Step 1: Recognize the Conic Section

The given polar equation is of the form \( r = \frac{ed}{1 + e \sin \theta} \) which represents a conic section with the eccentricity \( e \) given by the form \( r = \frac{5}{2 + 2 \sin \theta} \). From here, \( e = 1 \), indicating that this is a parabola.

Step 2: Identify the Direction and Directrix

For a parabola in polar form \( r = \frac{ed}{1 + e \sin \theta} \), the directrix lies parallel to the line \( \theta = 0 \). The parameter \( d = \frac{5}{2} \) is the distance from the focus to the directrix. Thus, the directrix is \( r = \frac{5}{2} \).

Step 3: Graph the Parabola and Directrix

Graph the parabola using its polar equation \( r = \frac{5}{2 + 2 \sin \theta} \). Also, plot the line \( r = \frac{5}{2} \) for all directions, which is the directrix of the parabola.

Step 4: Determine the Focus

For parabolas in this polar form, the focus is at the pole (the origin \( (0,0) \) in polar coordinates). This point remains unchanged even when rotating the parabola.

Step 5: Rotate the Parabola

A clockwise rotation by \( \frac{\pi}{6} \) (30 degrees) modifies the angle \( \theta \) to \( \theta - \frac{\pi}{6} \). The new equation becomes \( r = \frac{5}{2 + 2 \sin(\theta - \frac{\pi}{6})} \). Graph this rotated parabola.

Key Concepts

Conic Sections

Conic sections are fascinating curves obtained by slicing a double-napped cone. These sections include circles, ellipses, parabolas, and hyperbolas. Each shape has unique properties based on how the cone is intersected.
An essential parameter involved in defining conic sections is the eccentricity, denoted by the symbol \( e \). This value helps determine the type of conic:

• For a circle, \( e = 0 \).
• An ellipse has \( e < 1 \).
• A parabola occurs when \( e = 1 \).
• And a hyperbola forms when \( e > 1 \).

Understanding these basics is crucial as conic sections frequently appear in various mathematical contexts, such as algebra, calculus, and even physics. When dealing with these curves, recognizing their algebraic properties and their geometric implications can tremendously benefit your comprehension of their behavior.

Parabolas

Parabolas are a specific type of conic section characterized by their unique U-shaped curve. One of the interesting things about parabolas is that they have an eccentricity \( e = 1 \), which means they are the dividing line between ellipses and hyperbolas.
In a typical Cartesian plane, a parabola may look like \( y = ax^2 + bx + c \). However, in polar coordinates, it transforms into a different equation that reveals more about its geometric properties.
The standard polar form of a parabola is given by: \( r = \frac{ed}{1 + e \sin \theta} \), where:

• \( d \) is the perpendicular distance from the directrix to the focus.
• At \( e = 1 \), this equation simplifies specifically for parabolas.

By understanding its equation in polar form, you can plot it with respect to a point of focus, enhancing your grasp of how parabolas behave under transformation or rotation in space.

Graphing Polar Equations

Graphing polar equations involves plotting points in a coordinate system where each point is defined by a distance from the origin and an angle from the positive x-axis. This system differs from Cartesian coordinates and is particularly useful for equations that naturally incorporate angles and radii.
When attempting to graph, consider the following steps:

• Identify the form of the polar equation, in this case, a conic section.
• Determine key elements such as the directrix or axis of symmetry, if applicable.
• Convert angles as needed, such as when rotating the graph.

After identifying all elements of the polar equation, you can begin plotting by substituting various values of \( \theta \) to find corresponding \( r \) values.
Employing a combination of understanding the equation's form and utilizing polar graphing tools allows you to accurately visualize the behavior and traits of intricate curves, like the parabola, when plotted in polar coordinates.