Problem 33 - Precalculus Mathematics for Calculus

Question

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{6}{2+\sin \theta}$$

Step-by-Step Solution

Answer

Eccentricity is 1, identifying the conic as a parabola.

Step 1: Identify the Conic Form

The given equation is in the form \( r = \frac{ed}{1 + e \sin \theta} \), where \( e \) is the eccentricity and \( d \) is a constant related to the directrix. By comparing, here \( ed = 6 \) and \( 1 + e = 2 \) (since the form shows \( 2 + \sin \theta \)).

Step 2: Solve for Eccentricity

From the equation in Step 1, we have \( 1 + e = 2 \). Solving for \( e \), we find that \( e = 1 \).

Step 3: Identify the Conic Type

An eccentricity \( e = 1 \) indicates that the conic section is a parabola. Parabolas have \( e = 1 \), ellipses have \( 0 < e < 1 \), and hyperbolas have \( e > 1 \).

Step 4: Sketch the Conic

A parabola with the equation \( r = \frac{6}{2 + \sin \theta} \) means it opens around the focus described by changes in \( \theta \). To sketch, draw a rough parabola that opens upwards with its vertex at the pole.

Step 5: Label the Vertices

For a parabola in polar coordinates centered at the pole, the vertex is at the origin, \((0,0)\), relative to the focus and the directrix identified from the form of the equation.

Key Concepts

Eccentricity

Eccentricity is a fundamental concept that helps us identify and classify conic sections. It is a simple number denoting how "stretched" a conic section is. The value of eccentricity, denoted as \( e \), varies for different conics:

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), we have an ellipse.
• If \( e = 1 \), the conic is a parabola.
• If \( e > 1 \), the conic forms a hyperbola.

In this exercise, we determined that \( e = 1 \), which characterizes the conic as a parabola. Eccentricity is crucial because it not only tells us the type of conic but also guides us in sketching and understanding its properties better. Remember, the eccentricity gives you a snapshot comparison between different conics. Think of it as how much it deviates from being circular.

Polar Coordinates

Polar coordinates offer a unique way to describe the location of a point in a plane. Instead of using the traditional \((x, y)\) Cartesian coordinates, polar coordinates use a combination of distance \( r \) from a fixed point (called the pole, similar to the origin in Cartesian coordinates) and an angle \( \theta \), measured from a fixed direction (often the positive \( x \)-axis in Cartesian systems).

• \( r \): Distance from the pole.
• \( \theta \): Angle from the fixed direction.

In dealing with conics like parabolas, polar coordinates allow us to express equations conveniently, as seen in the polar equation \( r = \frac{6}{2 + \sin \theta} \). This approach often simplifies finding the nature and properties of the conic sections, such as the direction it opens towards and how it can be sketched. Polar coordinates can sometimes make complex equations look simpler by focusing on angles and distances directly.

Parabola

A parabola is a distinct conic section represented by a particular set of equations. When a parabola is expressed in polar coordinates, it offers a unique perspective that can be advantageous for understanding its geometry. The characteristic property of a parabola is that it has an eccentricity \( e = 1 \). This means it is neither closed like an ellipse nor open like a hyperbola, but perfectly arches between extremes.

• Parabolas have a single vertex, which can be considered the "sharpest" point.
• They are symmetric, meaning both halves mirror each other from the vertex.

In this problem, the parabola equation \( r = \frac{6}{2 + \sin \theta} \) opens upwards relative to its center. Such polar forms help clarify how the parabola will appear when sketched—the parabola stretches out from a point where \( \theta \) changes, creating its distinct shape. Understanding a parabola's polar form involves recognizing how it is defined by the angle \( \theta \) and how it revolves around the focus.