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

The conic is a parabola with eccentricity 1.

Step 1: Identify the Conic Type

The given polar equation is of the form \( r = \frac{ed}{1 + e\sin(\theta)} \). This is a standard form of a conic section in polar coordinates where the conic is described in terms of its eccentricity \( e \). We can compare this to our given equation: \( r = \frac{6}{2 + \sin(\theta)} \). By comparing, we find that \( e = 1 \) and \( ed = 6 \). Thus, \( d = 6 \).

Step 2: Calculate Eccentricity

From the provided equation, we have determined that \( e = 1 \). This eccentricity indicates that the conic is a parabola, because a parabola is characterized by an eccentricity \( e = 1 \).

Step 3: Confirm the Conic Section

With \( e = 1 \), we verify that the conic section is indeed a parabola. Parabolas have the property that their eccentricity is always 1.

Step 4: Sketching the Conic

To sketch the conic, we note that the equation \( r = \frac{6}{2 + \sin\theta} \) is symmetric concerning the horizontal axis. The vertex is at the minimum value of \( r \). By setting \( \sin\theta = -1 \), we find the minimum value of \( r = \frac{6}{1} = 6 \). The vertex of the parabola is thus at \( \theta = 270° \) or \( \theta = \frac{3\pi}{2} \) radians, at \( r = 6 \).

Key Concepts

Eccentricity

Eccentricity is a fundamental concept when studying conic sections. It is a number that describes the shape of a conic. This attribute defines how much a conic deviates from being circular.
For various conic sections, eccentricity (\( e \)) plays a crucial role:

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), the conic is an ellipse.
• If \( e = 1 \), as in our discussed exercise, the conic is a parabola.
• If \( e > 1 \), it forms a hyperbola.

In our original problem, the given polar equation is \( r = \frac{6}{2 + \sin \theta} \). On comparison with the standard form \( r = \frac{ed}{1 + e\sin(\theta)} \), we discovered that \( e = 1 \), which indicates it is a parabola. Eccentricity perfectly dictates the behavior and properties of the conic involved.

Polar Coordinates

The Polar Coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance and an angle. This system is particularly useful when dealing with curves like spirals and conic sections.
In polar coordinates:

• "r" denotes the radial distance from the origin.
• "𝜃" represents the angle from the positive x-axis.

The given exercise uses polar coordinates to express the conic section as \( r = \frac{6}{2 + \sin \theta} \). Here, for any given angle \( \theta \), the radius \( r \) changes based on \( \sin \theta \), illustrating the position of the conic's points. Understanding polar equations helps us visualize and sketch the conic sections effectively on a polar grid.

Parabola

A parabola is a specific type of conic section characterized by its eccentricity of \( e = 1 \). Unlike ellipses or hyperbolas, a parabola is open and extends infinitely in one direction.
This conic can be described and visualized in both Cartesian and polar coordinates. In the polar coordinate form, as given in our exercise \( r = \frac{6}{2 + \sin \theta} \), the parabola is defined by its distance equation relative to a focus point.
To sketch a parabola:

• Identify the vertex, which is the minimum or maximum value of \( r \). Here, it's at \( \theta = 270° \)
• Note the symmetry about the principal axis, guiding its extension.
• Recognize its open nature that doesn't enclose any space.

In our exercise, setting \( \sin \theta = -1 \) provides the vertex at the minimum \( r \) value, showcasing the parabolic path.