Problem 7 - Calculus Early Transcendentals

Question

Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity \(\frac{1}{2}, \quad\) directrix \(r=4 \sec \theta\)

Step-by-Step Solution

Answer

The polar equation for the ellipse is \( r = \frac{4}{2 + \cos \theta} \).

Step 1: Understand the General Equation

For a conic section with a focus at the origin, the polar equation is \( r = \frac{ed}{1 + e \cos \theta} \), where \( e \) is the eccentricity and \( d \) is the distance from the directrix to the origin.

Step 2: Identify the Given Values

It's given that the conic is an ellipse with an eccentricity \( e = \frac{1}{2} \) and the directrix is given by the equation \( r = 4 \sec \theta \). This means \( d = 4 \).

Step 3: Plug Values into the Equation

Substitute \( e = \frac{1}{2} \) and \( d = 4 \) into the general polar equation. Thus, the polar equation becomes \( r = \frac{\frac{1}{2} \cdot 4}{1 + \frac{1}{2} \cos \theta} \).

Step 4: Simplify the Equation

Carry out the multiplication in the numerator of the equation: \( r = \frac{2}{1 + \frac{1}{2} \cos \theta} \). Simplify further by multiplying both the numerator and the denominator by 2 to eliminate the fraction in the denominator, resulting in: \( r = \frac{4}{2 + \cos \theta} \).

Key Concepts

Ellipse

An ellipse is a type of conic section that resembles an elongated circle. It appears when a plane intersects a cone at an angle less than that of the cone's side, but it doesn't pass through the base. In the context of polar coordinates, an ellipse can be described by specific equations that involve geometric properties like eccentricity and directrix.

In an ellipse, every point has a constant sum of distances from two fixed points called foci. When the focus is located at the origin, it's crucial to understand how the shape and size of the ellipse are determined by other factors, such as the eccentricity and the directrix. Understanding these properties will help in writing the polar equation of the ellipse effectively.

Eccentricity

Eccentricity, denoted as "e," is a measure of how much a conic section deviates from being circular. For ellipses, the eccentricity ranges between 0 and 1, where an eccentricity of 0 characterizes a perfect circle.

In the case of our exercise, the eccentricity is given as \(\frac{1}{2}\). This value indicates that the ellipse is more circular compared to more elongated ellipses, which would have eccentricities closer to 1. An ellipse with eccentricity \(\frac{1}{2}\) is moderately stretched out but still maintains a significant circular resemblance. Incorporating the eccentricity is essential for deriving the correct polar equation, as it plays a pivotal role in the shape's properties.

Directrix

The directrix in a conic section is a line that guides the shape of the curve along with the focus. In polar coordinates, it is usually expressed in a form involving \( \theta \). The directrix helps to define the relationship of points on the ellipse relative to the focus.

For the given problem, the directrix is noted with the equation \( r = 4 \sec \theta \). This relationship helps in describing how far each point on the ellipse is from the straight line directrix and offers a constant measure combined with the focus. In polar equations, the directrix's relationship to the focus and the eccentricity is what defines the behavior of the conic section, helping to derive the polar equation by plugging the provided values into the general equation form.

Focus at the Origin

The location of the focus is crucial in determining the nature of the conic section. Here, the focus is positioned at the origin, meaning it is at the coordinate point (0,0) in the polar system. This position simplifies the equations of conics considerably.

Having the focus at the origin allows for a straightforward application of the formula \( r = \frac{ed}{1 + e \cos \theta} \). This formula involves the focus at a central point, where the variable "r" measures the radial distance from the origin to the conic section, and "\(\theta\)" measures the angle. This reference point is key as it centers the ellipse with respect to the radial symmetry inherent in polar coordinates.