Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the type of conic represented by the polar equation and analyze its graph. Then use a graphing utility to graph the polar equation. $$r=\frac{5}{1-\sin \theta}$$

Step-by-Step Solution

Answer

The equation \(r=\frac{5}{1-\sin \theta}\) represents a limaçon.

Step 1: Identify the form of Conic

Compare the given equation \(r=\frac{5}{1-\sin \theta}\) to the general form \(r=\frac{p}{1+e\sin \theta}\), with a minus sign in place of a plus sign. The eccentricity \(e\) is represented by a value of -1. Since the eccentricity is not 1, the conic section is not a parabola.

Step 2: Determine Conic Type

For -1 < e < 1, the conic is an ellipse. For e = 1, the conic is a parabola. For e > 1, the conic is a hyperbola. Since here e < -1, the equation represents a limaçon.

Step 3: Graph the Equation

Utilize a graphing utility to draw the polar equation r=5/(1-sin(theta)). Be aware that the graph might look distorted because a polar graphing utility is in spherical coordinates. A polar coordinate graph can be visually verified as a limaçon.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone. They are fascinating because they include ellipses, parabolas, hyperbolas, and circles. Each type of conic section can be represented within the polar coordinate system using specific equations.
In a polar equation, the form typically used is \(r = \frac{p}{1 + e\sin \theta}\) or \(r = \frac{p}{1 + e\cos \theta}\). Here, \(p\) is a constant, \(e\) is the eccentricity, and \(\theta\) is the polar angle. The value of \(e\) determines the type of conic section:

• If \(e = 0\), the conic is a circle.
• If \(0 < e < 1\), it forms an ellipse.
• If \(e = 1\), it is a parabola.
• If \(e > 1\), it becomes a hyperbola.

Identifying the conic type involves comparing the given equation to this standard form.

Limaçon

A limaçon is a type of polar graph that resembles a distorted circle and belongs to the broader family of conic sections. The term "limaçon" comes from the French word for "snail," due to its shell-like shape. It can be defined using an equation in polar coordinates, typically represented as \(r = a + b\sin \theta\) or \(r = a + b\cos \theta\).
The shape of a limaçon changes based on the ratio of \(a\) to \(b\):

• If \(|a| = |b|\), the limaçon has an inner loop.
• If \(|a| > |b|\), it forms a dimpled or cardioid shape.
• If \(|a| < |b|\), an outer loop exists.

Understanding the shape and properties of a limaçon involves recognizing its dependence on polar coordinates and the specific values in its equation.

Eccentricity

Eccentricity is a critical concept in understanding conic sections. It describes how much a conic section deviates from being circular. In the context of polar equations, it is denoted as \(e\) and appears in the standard forms like \(r = \frac{p}{1 + e\sin \theta}\).
The value of eccentricity gives us vital information:

• \(e = 0\) indicates a perfect circle.
• \(0 < e < 1\) suggests an ellipse.
• \(e = 1\) signifies a parabola, highlighting a special symmetrical curve.
• \(e > 1\) means a hyperbola, showing a split curve.

Eccentricity is crucial for identifying the type and nature of a conic, as seen when analyzing polar equations.

Graphing Utility

A graphing utility is a powerful tool for visualizing complex mathematical equations, especially those expressed in polar coordinates. These tools help bring equations to life, making it easier to understand their shapes and properties.
When graphing the equation \(r = \frac{5}{1 - \sin \theta}\), a graphing utility can assist by plotting the points and illustrating the curve’s full form. It can differentiate between standard conic sections and more unique formations like limaçons.
Using a graphing utility comes with several benefits:

• Accurate visual representation of equations.
• Dynamic interaction, allowing for changes and experimentation.
• Immediate feedback, helping to reinforce mathematical concepts.

They transform abstract concepts into tangible visuals, perfect for any student tackling polar equations.