Problem 19 - Precalculus: Mathematical 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 0.5; the conic is an ellipse.

Step 1: Identify the Conic Type

Start by writing the equation in the standard polar form \( r = \frac{ed}{1 + e \sin \theta} \). Here, \( r = \frac{6}{2 + \sin \theta} \) can be rewritten as \( r = \frac{6}{2 (1 + \frac{1}{2} \sin \theta)} \). Thus, comparing with the standard form, \( e = \frac{1}{2} \) and \( d = 3 \).

Step 2: Determine the Eccentricity

The eccentricity \( e \) is the coefficient of \( \sin \theta \) in the denominator compared to 1. From the equation, \( e = \frac{1}{2} \).

Step 3: Identify the Conic Section

The eccentricity \( e = \frac{1}{2} \) indicates an ellipse because for ellipses, \( 0 \leq e < 1 \).

Step 4: Sketch the Conic

An ellipse in polar coordinates centered at the pole will have vertices at the maximum and minimum values of \( r \). The minimum \( r \) occurs when \( \sin \theta = 1 \), yielding \( r_{min} = \frac{6}{2+1} = 2 \). The maximum \( r \) occurs at \( \sin \theta = -1 \), giving \( r_{max} = \frac{6}{2-1} = 6 \).

Step 5: Label the Vertices

For \( \theta = \frac{\pi}{2} \), \( r = 2 \), and for \( \theta = \frac{3\pi}{2} \), \( r = 6 \). Thus, the vertices of the ellipse are \((2, \frac{\pi}{2})\) and \((6, \frac{3\pi}{2})\).

Key Concepts

Eccentricity

Eccentricity is a fundamental concept used to describe the shape of conic sections. It is a number that quantifies how much a conic section deviates from being circular. Generally, the eccentricity is denoted by the symbol \( e \).

The value of eccentricity determines the type of conic section:

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

In the exercise, we have determined that the eccentricity is \( e = \frac{1}{2} \), indicating the conic is an ellipse. This means the shape is more elongated than a circle but does not open like a parabola or hyperbola. Eccentricity serves as a key parameter in understanding how conic sections behave in geometry and other mathematical applications.

Ellipse

An ellipse is a type of conic section characterized by its oval shape. The key property of an ellipse is that it has two focal points. The sum of the distances from any point on the ellipse to these two foci is constant.

The standard equation for an ellipse in polar coordinates can be written as:\[r = \frac{ed}{1 + e \sin \theta}\]Here, \( e \) is the eccentricity, and \( d \) is the semi-latus rectum. In our exercise, this formula helps identify the type of conic section and provides insights into its geometry. We found the vertices, which are the nearest and furthest points from the origin to the conic, to be at \( r_{min} = 2 \) and \( r_{max} = 6 \).

Understanding ellipses is crucial because they appear in various scientific contexts, such as planetary orbits, where planets move in elliptical paths around stars. The balanced and symmetric nature of ellipses makes them not only a fundamental geometric object but also a model for complex systems in physics.

Polar Coordinates

Polar coordinates offer a unique way to describe the position of a point in a plane. Unlike the Cartesian coordinate system, which uses \( x \) and \( y \), polar coordinates express a point using a radius and an angle, \( (r, \theta) \).

In our exercise, the equation \( r = \frac{6}{2 + \sin \theta} \) describes an ellipse in terms of polar coordinates. Here, \( r \) represents the distance from the pole (origin), and \( \theta \) is the angle measured from the polar axis. This representation is especially useful for conic sections with rotational symmetry around the origin.

Using polar coordinates can simplify the analysis of certain geometric shapes and curves, especially when symmetry and central points are involved. This system allows for natural descriptions of motion or growth problems, such as oscillating or spiraling patterns, which often arise in physics and engineering contexts.