Problem 33 - Thomas Calculus

Question

Exercises \(29-36\) give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$e=1 / 2, \quad x=1$$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{1}{2 + \cos \theta} \).

Step 1: Understand the Components

In the given problem, we have the eccentricity of the conic section, which is \( e = \frac{1}{2} \), and the equation of the directrix, which is \( x = 1 \). A conic section in polar coordinates with its focus at the origin can be defined using the polar equation: \( r = \frac{ed}{1 + e \cos \theta} \), where \( e \) is the eccentricity, and \( d \) is the distance from the focus to the directrix.

Step 2: Determine the Values

Since the directrix is \( x = 1 \), for a horizontal directrix \( d = 1 \). The eccentricity is given as \( e = \frac{1}{2} \).

Step 3: Substitute Values into the Polar Equation

Substitute the values of \( e \) and \( d \) into the polar equation \( r = \frac{ed}{1 + e \cos \theta} \) to get:\[r = \frac{\frac{1}{2}\cdot 1}{1 + \frac{1}{2} \cos \theta} = \frac{\frac{1}{2}}{1 + \frac{1}{2} \cos \theta} \]

Step 4: Simplify the Polar Equation

Simplify the expression for \( r \): Multiply numerator and denominator by 2 to clear the fraction:\[r = \frac{1}{2} \cdot \frac{2}{2} \frac{1}{1 + \frac{1}{2}\cos \theta} = \frac{1}{2 + \cos \theta}\]Thus, the polar equation of the conic section is \( r = \frac{1}{2 + \cos \theta} \).

Key Concepts

Eccentricity in Conic Sections

Eccentricity is a crucial parameter in understanding conic sections. It determines the shape of the conic section. Conic sections can be:

• Ellipses if eccentricity, denoted as \( e \), is between 0 and 1.
• Parabolas if \( e = 1 \).
• Hyperbolas if \( e > 1 \).

In our problem, the eccentricity is \( e = \frac{1}{2} \). This value indicates that the conic section is an ellipse as it lies between 0 and 1.
The closer the eccentricity is to 0, the more the ellipse resembles a circle. As it approaches 1, the ellipse becomes more elongated. This property makes eccentricity a fundamental part of identifying and describing the general appearance of conic sections.

Polar Coordinates

Polar coordinates offer a unique way to describe points in a plane using the distance from a fixed point, known as the pole (usually the origin), and the angle from a fixed direction, typically the positive x-axis.
This system is especially useful for describing curves and shapes centered around a point, like our conic section.
In the context of conic sections, the polar equation is generally given by:\[ r = \frac{ed}{1 + e \cos \theta} \]Here:

• \( r \) is the radial distance from the pole to a point on the curve.
• \( \theta \) is the angle from the polar axis.
• \( e \) is the eccentricity.
• \( d \) is the distance from the focus to the directrix.

This form provides an easy method to visualize and understand how a conic curve behaves with respect to the pole in polar coordinates. It is the backbone for solving problems involving the shapes formed by these equations.

Directrix of Conic Sections

The directrix is a fixed line used in the definition of conic sections. For any point on a conic section, the ratio of its distance from a focus to its distance from the directrix is constant and is equal to the eccentricity \( e \).
When working with polar equations, the distance \( d \) from the focus to the directrix is pivotal in determining the shape of the conic.
In this exercise, the directrix is given as the vertical line \( x = 1 \). This specific configuration suggests:

• A horizontally aligned conic section.
• The directrix influences the form of the polar equation \( r = \frac{ed}{1 + e \cos \theta} \).

It signifies that by knowing the position of the directrix and the eccentricity, we can derive the equation to depict the conic effectively in a coordinate plane.