Problem 19 - Algebra and Trigonometry

Question

15–22 (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 \( e = 1 \); it is a parabola.

Step 1: Identify the Conic Form

The given equation is \( r = \frac{6}{2 + \sin \theta} \). This resembles the form \( r = \frac{ed}{1 + e\sin\theta} \), which indicates a conic section.

Step 2: Determine the Eccentricity and Parameters

We can rewrite the equation as \( r = \frac{6}{2 + 1 \cdot \sin \theta} \). Comparing with \( r = \frac{ed}{1 + e\sin\theta} \), we find \( ed = 6 \) and \( 1 + e = 2 \). Thus, \( e = 1 \).

Step 3: Identify the Type of Conic

Since the eccentricity \( e = 1 \), this conic is a parabola. A parabola has an eccentricity exactly equal to 1.

Step 4: Sketch the Parabola

The equation can be rewritten as \( r = \frac{6}{1} \) when \( \theta = \frac{\pi}{2} \). Solve for the vertex, i.e., \( r = 3 \) for \( \theta = \frac{\pi}{2} \), hence the vertex is at polar coordinates \((3, \frac{\pi}{2})\). Plot this and note the parabola extends upwards, opening in the direction of the sine term.

Key Concepts

Eccentricity

Eccentricity is a fundamental measure used to describe the shape of a conic section. The eccentricity value, often denoted as \( e \), dictates whether the conic section is a circle, ellipse, parabola, or hyperbola.
To determine the type of conic, observe the value of \( e \):

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), it's an ellipse.
• If \( e = 1 \), the conic is a parabola.
• If \( e > 1 \), it's a hyperbola.

In the context of this exercise, we examined the equation \( r = \frac{6}{2+\sin\theta} \) and found that the eccentricity \( e = 1 \). This places our conic as a parabola, which is defined by having an eccentricity exactly equal to 1.
This key feature helps distinguish parabolas from other conics, making it essential to solving problems involving conic sections.

Parabola

A parabola is a unique conic section that you encounter when the eccentricity \( e \) is equal to 1. Unlike ellipses and hyperbolas, parabolas have a distinct geometric property: their paths are symmetrical, resembling an open curve with a single vertex.
The equation \( r = \frac{ed}{1 + e\sin\theta} \) was compared to \( r=\frac{6}{2+\sin\theta} \), illustrating that for \( e = 1 \), the conic is indeed a parabola.

The vertex of this parabola in polar coordinates was found to be \((3, \frac{\pi}{2})\), which serves as the peak or tip of the parabola. This represents the closest point on the parabola to the focus, found at the origin in polar plots. The parabola in this scenario opens upwards, complying with the positive sine component in the equation.
Understanding the parabola's structure aids in sketching and pinpointing other features like the directrix, a line equidistant from the parabola as the focus, situated on the opposite side of the vertex.

Polar Coordinates

Polar coordinates offer a unique way of representing points in a plane using a pair of values: the radial distance \( r \) from a fixed point (origin) and the angular coordinate \( \theta \), the angle measured from the reference direction (typically the positive x-axis).
This system is particularly powerful for describing curves such as conic sections that may have branches or symmetry about a point, as seen in this exercise involving a parabola.

The given polar equation \( r = \frac{6}{2 + \sin \theta} \) shows the relationship between \( r \) and \( \theta \), defining the conic's shape and position, which in this case is a parabola. As \( \theta \) varies, \( r \) will describe the point's distance from the origin, revealing the curve's extent and orientation.
Working through polar coordinates, you can sketch the curve by finding various points for different angles \( \theta \), assisting in visualizing how the curve behaves and determining features such as the conic's vertex.