Problem 8 - Precalculus Mathematics for Calculus

Question

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(0.6,\) directrix \(r=2 \csc \theta\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{1.2}{1 - 0.6 \sin \theta} \).

Step 1: Understanding the Given Conditions

We are given an ellipse with an eccentricity of \( e = 0.6 \) and a directrix described by the equation \( r = 2 \csc \theta \). The focus of this conic is at the origin.

Step 2: Formula for Polar Equation of a Conic

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

Step 3: Substituting Given Eccentricity and Directrix

Given the directrix is \( r = 2 \csc \theta \), this implies \( d = 2 \) because the csc function implies that the distance from the directrix to the pole (origin) is 2. Given \( e = 0.6 \), substituting into the formula: \[ r = \frac{0.6 \times 2}{1 - 0.6 \sin \theta} \]

Step 4: Simplifying the Equation

Simplify the equation by multiplying: \[ r = \frac{1.2}{1 - 0.6 \sin \theta} \]This is the polar equation of the ellipse that satisfies all given conditions.

Key Concepts

Conic Sections

Conic sections are shapes created as a plane cuts through a double-napped cone. Imagine slicing a cone in different ways—each slice reveals a different shape which we call a conic section. These sections include circles, ellipses, parabolas, and hyperbolas.

• A circle is when the slice is parallel to the base of the cone.
• An ellipse results from a slice at an angle, but not steep enough to intersect both nappes of the cone.
• A parabola forms when the slice is parallel to the slope of the cone.
• A hyperbola is produced when the slice intersects both pieces of the cone.

In this exercise, the conic section in question is an ellipse. Ellipses are characterized by their oval shape, where they have two foci and any point on the ellipse is the sum of the distances to these foci. In polar coordinates, the focus of the ellipse we’re dealing with is at the origin, making the equation simple and dependent on eccentricity and directrix.

Eccentricity

Eccentricity is a measure of how much a conic section deviates from being circular. It's a ratio that defines the shape's elongation.

• For a circle, the eccentricity is 0, as it’s perfectly round.
• In an ellipse, the eccentricity ranges between 0 and 1. The closer it is to 1, the more elongated the ellipse is.
• A parabola has an eccentricity of exactly 1.
• In a hyperbola, the eccentricity is greater than 1, indicating it is more stretched out.

In this particular problem, our ellipse has an eccentricity of 0.6. This tells us that the ellipse is not a perfect circle but isn't too stretched out either. It confers a moderate elongation typical of ellipses with foci that lie closer together compared to wider ones.

Directrix

A directrix is a line used to define a conic section, aiding in its construction along with the eccentricity.

• For ellipses and hyperbolas, the directrix helps in positioning the curve in relation to its focus.
• A parabola uses a directrix to define every point on it as equidistant from the focus as from the directrix.
• Directrices can be parallel to each other in multiple-branch conic sections like hyperbolas.

In polar coordinates, the directrix often helps in establishing the polar equation of the conic. Here, it's noted as \( r = 2 \csc \theta \), representing a straight line. This tells us that the distance from any point on the ellipse to the line is involved in defining the shape of the ellipse. The parameter \( d = 2 \) from the directrix equation is used synonymously with eccentricity to form the polar equation.