Problem 43 - College Algebra

Question

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=4 ; e=\frac{1}{5}\)

Step-by-Step Solution

Answer

The polar equation of the conic is \( r = \frac{4}{5 + \cos\theta} \).

Step 1: Identify Type of Conic

Given the eccentricity \(e = \frac{1}{5}\), we have \(0 < e < 1\), which indicates that the conic is an ellipse.

Step 2: Use the Polar Equation Definition

For a conic with eccentricity \(e\), focus at the origin, and directrix \(x = d\), the polar equation is given by:\[ r = \frac{ed}{1 + e\cos\theta} \] where the directrix \(x = d\) implies \(d = 4\).

Step 3: Substitute Known Values

Substitute \(e = \frac{1}{5}\) and \(d = 4\) into the polar equation formula:\[ r = \frac{\left(\frac{1}{5}\right)\cdot 4}{1 + \frac{1}{5}\cos\theta} \]

Step 4: Simplify the Equation

Simplify the expression:\[ r = \frac{\frac{4}{5}}{1 + \frac{1}{5}\cos\theta} = \frac{4}{5(1 + \frac{1}{5}\cos\theta)} = \frac{4}{5 + \cos\theta} \]

Step 5: Final Polar Equation of the Conic

The final polar equation of the conic is:\[ r = \frac{4}{5 + \cos\theta} \]

Key Concepts

Ellipse

An ellipse is a fascinating conic section that arises when a plane cuts through a cone at an angle smaller than that of the side of the cone. It has an oval shape and is defined by two points known as foci. The sum of the distances from any point on the ellipse to the two foci is always constant.
When we consider conic sections in a polar coordinate system, ellipses appear when the eccentricity, often denoted as "e," is greater than 0 but less than 1. This range of eccentricity ensures that the shape remains oval instead of becoming more circular or parabola-like.
With polar equations, we start with a reference point at the origin (focus) and use the relationship between distance to the focus and a specific line (directrix) to define the ellipse. An understanding of eccentricity is critical, as it helps in distinguishing ellipses from other conic sections such as parabolas and hyperbolas.

Eccentricity

Eccentricity provides a quantitative measure of how much a conic section deviates from being circular. It plays a crucial role in determining the shape of conic sections, including ellipses, parabolas, and hyperbolas.
For an ellipse, the eccentricity is denoted by a number "e" that lies between 0 and 1. The closer "e" is to 0, the more circular the ellipse. Conversely, as "e" gets closer to 1, the ellipse becomes more elongated.

• For any eccentricity value "e" such that 0 < e < 1, we indeed have an ellipse.
• In our case, e = \( \frac{1}{5} \), showing that the figure is an ellipse, as it fits within the eccentricity range for ellipses.

The concept of eccentricity helps us compare different conic sections and easily identify the nature of a section given its geometric properties.

Directrix

A directrix is a reference line used to measure distances in the description of conic sections. Its primary role is to assist in defining the shape and position of a conic with respect to a focus.
For ellipses and other conics, the ratio of the distance from any given point on the conic to a focus over the distance from that point to the directrix is constant. This constant ratio is the eccentricity "e."

• In polar coordinates, for a conic with focus at the origin, the directrix is usually set at a certain distance "d" from the origin.
• In this exercise, the directrix is given as \(x = 4\), which assists in forming the polar equation by relating distances in a unique and consistent manner.

The directrix complements the focus in uniquely defining the conic's equation and ensuring it adheres to its required geometric properties.

Polar Coordinates

Polar coordinates provide a way to describe locations on a plane using a radius and an angle, rather than traditional Cartesian coordinates (x, y). They simplify the representation of conic sections, especially when a focus is at the origin.
In the polar coordinate system, each point is described by \( (r, \theta) \), where "r" is the distance from the origin to the point, and "θ" is the angle formed with the positive x-axis.
When dealing with conic sections, the focus of the conic sits at the origin (the pole in polar coordinates). The given directrix and eccentricity determine the specific polar equation:

• This relationship is shown in the equation \[ r = \frac{ed}{1 + e \cos \theta} \], which translates the properties of the conic in a compact, easy-to-use format.
• It helps visualize and solve problems related to conic sections by focusing on the direct relationship between distance and angle.

Polar coordinates transform the approach to solving and understanding conic sections, promoting accuracy and simplicity in mathematical descriptions.