Problem 39 - Precalculus with Limits

Question

In Exercises 39-54, find a polar equation of the conic with its focus at the pole. \(\textit{Conic}\) Parabola \(\textit{Eccentricity}\) \(e=1\) \(\textit{Directrix}\) \(x=-1\)

Step-by-Step Solution

Answer

The polar equation of the parabola is \(r = \frac{1}{1 - \cos(\theta)}\).

Step 1: Understand the standard form of a parabola

A parabola with focus at the origin (the pole in polar coordinates) and directrix \(x = -p\) (or in terms of polar coordinates, \(r \cos(\theta) = -p\)) can be represented in polar coordinates by the equation \(r = \frac{e}{1 - e \cos(\theta)}\), where \(e\) is the eccentricity.

Step 2: Apply the given information

The given directrix is at \(x = -1\). Because the directrix is \(x = -p\) and perpendicular to the polar axis, we have \(p = 1\). Also, the given eccentricity \(e = 1\). Substituting these values into the equation, we get \(r = \frac{1}{1 - \cos(\theta)}\).

Step 3: Simplify the equation

The equation \(r = \frac{1}{1 - \cos(\theta)}\) represents the polar form of the parabola with a focus at the origin and a directrix at \(x = -1\). The equation cannot be simplified any further, so this is the final answer.

Key Concepts

Understanding the Parabola

A parabola is a unique type of conic section created when a plane intersects a cone parallel to its side. In simpler terms, imagine slicing a cone along its side; the curve it creates is a parabola. Parabolas have a distinct U-shape and have properties that make them interesting and useful in both mathematics and real-world applications, such as the paths of projectiles or in satellite dishes.

When we discuss parabolas in polar coordinates, their behavior changes slightly. A parabola in polar form is defined differently than in Cartesian coordinates. The focus of the parabola is often located at the origin, which is referred to as the pole in a polar coordinate system. The position of the parabola changes based on the location of the directrix and the parabola's eccentricity, both pivotal aspects of its equation.

Eccentricity of a Parabola

Eccentricity is a critical parameter that helps define the shape of a conic section, such as a circle, ellipse, or parabola. It shows how much a conic section deviates from being circular. The eccentricity, denoted by the symbol \( e \), is a numeric value that gives insight into the nature of the curve.

For a parabola, this value is always 1. This is an essential characteristic because it distinguishes parabolas from other conic sections.

• When \(e=0\), the conic is a circle.
• For \(e\) between 0 and 1, it forms an ellipse.
• At \(e=1\), we have a parabola.
• If \(e\) is greater than 1, the shape becomes a hyperbola.

The value \(e=1\) implies that the parabolic curve is equidistant from its focus and directrix at any point along its length. Understanding eccentricity allows you to predict and explain the parabola's shape and behavior.

The Role of the Directrix

The directrix is an imaginary line that plays a crucial role in the definition and formation of a conic section like a parabola. In the simplest terms, a parabola is any point equidistant from a focus and a line called the directrix.

The directrix helps determine the position and orientation of the parabola. In polar coordinates, the equation of the parabola involves the directrix, which is often given by the formula \(r = \frac{e}{1 - e \cos(\theta)}\). This equation outlines how each point on the parabola is related to both the focus and the directrix.

In our given exercise, the directrix is designated at \(x = -1\). This particular alignment changes the trajectory and ensures that the parabola faces in a specific direction. The importance of the directrix thus lies in its impact on the shape and orientation, critical for defining the parabola's equation in polar form.