Problem 29 - Precalculus: Functions and Graphs

Question

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$e=1, r \sin \theta=-2$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Given Information

We are given the eccentricity \( e = 1 \), indicating that this is a parabola, as parabolas have eccentricity equal to 1. We are also given the equation of the directrix as \( r \sin \theta = -2 \). This tells us the directrix is horizontal and located 2 units below the pole.

Step 2: Identify the Directrix Form

The equation \( r \sin \theta = -2 \) is in the form of \( r = \frac{d}{\sin \theta} \), where \( d = -2 \). The negative sign indicates that the directrix is below the pole on the polar axis.

Step 3: Polar Equation of a Conic

The general polar equation of a conic section with focus at the pole and directrix \( r = \frac{d}{\sin \theta} \) is given by:\[ r = \frac{ed}{1 + e \sin \theta}\] where \( e \) is the eccentricity and \( d \) is the distance to the directrix.

Step 4: Substitute Values into the Equation

Substitute \( e = 1 \) and \( d = -2 \) into the polar equation formula:\[ r = \frac{1 \cdot (-2)}{1 + 1 \sin \theta} = \frac{-2}{1 + \sin \theta}\]This represents the polar equation of the conic.

Key Concepts

Understanding Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. They include ellipses, parabolas, and hyperbolas. Each type of conic section has unique properties and forms based on its eccentricity. In the context of polar equations, conic sections play a crucial role in representing mathematical relationships in various fields.

• Ellipse: An ellipse is formed when the eccentricity is between 0 and 1. It looks like a stretched circle.


• Parabola: When the eccentricity is exactly 1, the conic section is a parabola. Parabolas are symmetric and have a singular curved shape.


• Hyperbola: A hyperbola is formed with an eccentricity greater than 1 and consists of two disconnected curves mirroring each other.

Understanding these shapes helps to solve problems involving polar equations by providing a visual reference for the mathematical descriptions.

The Role of Eccentricity in Conics

Eccentricity (denoted as \( e \)) is a fundamental property that defines the shape of a conic section. It describes how elongated the curve is. For different conic sections, eccentricity takes on distinct values that help identify and differentiate them.

• Zero Eccentricity: Represents a perfect circle, as circles are a special type of ellipse with no elongation.


• Eccentricity of 1: This occurs in parabolas, indicating a balance between the distance to the focus and the directrix.


• Greater than 1: Indicates a hyperbola, which appears as two opposing curves.

In our initial exercise, \( e = 1 \), confirming the conic is a parabola. This clear classification allows for further precise calculations.

Directrix and Its Importance

The directrix of a conic section is a fixed line used in the geometric definition of the curve. It works together with the eccentricity and the focus to define the conic's shape. In polar equations, the position of the directrix significantly affects the resulting graph.

• Location: For a given conic, the directrix can be positioned above, below, or beside the focus depending on whether the problem is solved in polar or Cartesian coordinates.


• Equation: Being expressed in different forms, the directrix's polar equation form in our study \( r \sin \theta = -2 \) indicates a horizontal line.


• Significance: In the formula \( r = \frac{ed}{1 + e \sin \theta} \), both \( e \) (eccentricity) and \( d \) (the signed distance to the directrix) determine the characteristic equation of a conic section in polar form.

A correct understanding of the directrix is key to accurately plotting and solving equations of conic sections in polar form.