Problem 280 - Calculus Volume 3

Question

Consider the following polar equations of conics. Determine the eccentricity and identify the conic. $$ r=\frac{8}{2-\sin \theta} $$

Step-by-Step Solution

Answer

The conic is an ellipse with eccentricity \( \frac{1}{4} \).

Step 1: Identify the Standard Form

Recognize that the given equation is in the form \( r = \frac{ed}{1 - e\sin(\theta)} \), where \( d = 8 \) and \( 2 = ed \).

Step 2: Find the Eccentricity

Using the relationship from the standard form, \( 2 = ed \), and knowing that \( d = 8 \), solve for \( e \):\[ e = \frac{2}{8} = \frac{1}{4}.\]

Step 3: Identify the Conic Type

Since the eccentricity \( e < 1 \), the conic is an ellipse. For an ellipse, the value of \( e \) is between 0 and 1.

Key Concepts

Eccentricity

Eccentricity is a key concept in the world of conic sections. It's a numerical value that describes the shape of a conic section. The eccentricity (\( e \) ) determines how much the conic section deviates from being circular.

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

In the exercise, the given polar equation \( r = \frac{8}{2 - \sin \theta} \) was identified as having an eccentricity of \( \frac{1}{4} \) , which is less than 1. This directly leads us to classify the conic as an ellipse, adhering to the typical properties of eccentricity.

Conic Sections

Conic sections are the result of intersecting a plane with a double-napped cone. Depending on how the plane cuts through the cone, different shapes, or conics, are formed.

• Circles and ellipses appear when the plane's angle is less steep than that of the cone's side.
• Parabolas form when the plane is parallel to the generatrix of the cone.
• Hyperbolas occur when the plane cuts through both nappes of the cone.

Each conic section has a different set of properties and is described by its own specific eccentricity value. This value is very useful in determining the type of conic section represented by a given equation. It's crucial in our problem-solving to recognize the relationship between the plane and the cone's angle, which directly links to the eccentricity and dictates the conic's nature.

Ellipse

An ellipse is one of the most common types of conic sections, characterized by its oval shape. The key property of an ellipse is that it has two focal points, and the sum of the distances from any point on the ellipse to the two foci is constant. This creates its unique, stretched circle form.

• An ellipse is defined by both its major and minor axes, which are the longest and shortest diameters, respectively.
• The eccentricity of an ellipse is always between 0 and 1, making it less circular as \( e \) approaches 1.
• The closer the eccentricity is to 0, the more the ellipse resembles a circle.

Polar equations are a useful way to describe ellipses in terms of radius and angle. The exercise we explored showed an equation with \( e = \frac{1}{4} \) , clearly identifying the conic section as an ellipse. Understanding the shape and properties of an ellipse is essential for connecting eccentricity values to real-world geometry applications.