Problem 344 - Calculus Volume 3

Question

Determine the eccentricity and identify the conic. Sketch the conic. $$ r=\frac{6}{1+3 \cos (\theta)} $$

Step-by-Step Solution

Answer

Eccentricity is 3; the conic is a hyperbola.

Step 1: Identifying the conic form

The given equation is \( r = \frac{6}{1 + 3 \cos(\theta)} \). This is of the form \( r = \frac{ed}{1 + e \cos(\theta)} \), which represents a conic section in polar coordinates with eccentricity \( e \).

Step 2: Determine eccentricity

Compare the equation \( r = \frac{6}{1 + 3 \cos(\theta)} \) with the standard form \( r = \frac{ed}{1 + e \cos(\theta)} \). Here, \( e = 3 \).

Step 3: Identify the conic section

Since the eccentricity \( e = 3 > 1 \), the conic is a hyperbola. Conic sections are hyperbolas when \( e > 1 \).

Step 4: Sketch the conic

To sketch the hyperbola in polar coordinates, note that a large eccentricity indicates the hyperbola opens widely. Plot key points for \( \theta = 0, \pi/2, \pi, 3\pi/2 \), considering how \( e = 3 \) affects the distance from the origin (focus) to the conic at these angles.

Key Concepts

Eccentricity

Eccentricity is a fundamental parameter that defines the shape and nature of conic sections. In simple terms, it measures how far a conic section deviates from being circular. The formula to determine eccentricity usually involves comparing it to certain values:

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

In our example, the equation provided is \( r = \frac{6}{1 + 3 \cos(\theta)} \). We determine the eccentricity by comparing this to the standard polar equation form: \( r = \frac{ed}{1 + e \cos(\theta)} \). Here, we can directly see that \( e = 3 \).
This value indicates a hyperbola, as \( 3 > 1 \). The greater the eccentricity, the more elongated or open the shape is, as is common with hyperbolas.

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system that specify the position of a point as a distance from a reference point and an angle from a reference direction. This system is particularly useful in scenarios involving curves and circles, where using regular Cartesian coordinates might lead to complicated equations.In polar coordinates, every point in a plane is identified with two values:

• - The radial distance \( r \), which is the distance from the pole (origin).
• - The angle \( \theta \), which is measured in radians from a fixed direction.

In the given problem, the equation \( r = \frac{6}{1 + 3 \cos(\theta)} \) was written in polar coordinates. This indicates relationships in a radial form, which can often simplify the classification and sketching of conic sections like hyperbolas. When sketching conic sections in polar coordinates, we typically vary \( \theta \) and compute the corresponding \( r \) to get various points that outline the graph.

Hyperbola

Hyperbolas are one of the four types of conic sections - the others being circles, ellipses, and parabolas. They are defined as the set of points where the difference of the distances to two fixed points, called foci, is constant. The distinguishing parameter for a hyperbola is its eccentricityHyperbolas have two separate branches, which are mirror images of each other and find their applications in fields like astronomy, radio transmission, and even in navigation technologies.
In our exercise, the hyperbola is defined by a condition \( e > 1 \). That's why, with an eccentricity \( e = 3 \), we correctly identify the graph as a hyperbola. Moreover, in the polar form \( r = \frac{6}{1 + 3 \cos(\theta)} \), the form of the equation ensures that as \( \theta \) varies from \(0\) to \(2\pi\), the polar curve will demonstrate a hyperbola that opens widely, making it straightforward to sketch key characteristics by plotting points at specific \(\theta\) values.