Question

For the following exercises, determine the eccentricity and identify the conic. Sketch the conic. $$ r=\frac{6}{1+3 \cos (\theta)} $$

Short answer

The conic is a hyperbola with eccentricity 3.

Step-by-step solution

Identify the General Form of the Polar Equation

The given equation is \( r = \frac{6}{1 + 3 \cos(\theta)} \). This equation can be compared to the standard form of a polar equation of a conic section: \( r = \frac{ed}{1 + e \cos(\theta)} \), where \( e \) is the eccentricity and \( d \) is a constant.

Determine the Eccentricity

From \( r = \frac{ed}{1 + e \cos(\theta)} \), equate \( ed = 6 \) and \( e = 3 \). Therefore, the eccentricity \( e \) is 3.

Classify the Conic Section

The value of the eccentricity \( e = 3 \) is greater than 1, which indicates that the conic section is a hyperbola.

Sketch the Conic

In polar coordinates, for \( e > 1 \) the shape should be a hyperbola. The center is at the pole, and one branch opens to the right, extending indefinitely, starting at the pole and moving outwards wherever \( \cos(\theta) \) equates to a value suitable for a real positive \( r \).

Key concepts

Eccentricity

Eccentricity is a key feature that characterizes different types of conic sections. In simple terms, it is a number that describes the shape of a conic section. The eccentricity determines whether a conic section is a circle, ellipse, parabola, or hyperbola.
- For circles, the eccentricity (\( e \)) is always 0. Circles have no flattening and are perfectly round.
- Ellipses have an eccentricity value between 0 and 1, indicating some degree of flattening.
- Parabolas have an eccentricity of exactly 1, representing their symmetrical, open shape.
- Lastly, hyperbolas, which are the focus of this exercise, have an eccentricity greater than 1.
In this exercise, when you see the equation \( r = \frac{6}{1 + 3 \cos(\theta)} \), by comparing it to the standard form \( r = \frac{ed}{1 + e \cos(\theta)} \), you find that the eccentricity (\( e \)) is 3. Since 3 is greater than 1, this confirms that the conic is a hyperbola. The measure of eccentricity essentially tells us how "stretched" the conic section is. So, when determining the nature of a conic, always check the value of the eccentricity.

Polar Coordinates

Polar coordinates offer an alternative way to describe the location of points in a plane, and they are especially useful in dealing with conical sections. Unlike Cartesian coordinates that use \( x \) and \( y \) to pinpoint locations, polar coordinates use a radius (\( r \)) and an angle (\( \theta \)).
- The radius (\( r \)) is the distance from a central point known as the pole, similar to the origin in Cartesian coordinates.
- The angle (\( \theta \)) is typically measured in radians from the positive \( x \)-axis, counter-clockwise around the plane.
In this problem, the equation \( r = \frac{6}{1 + 3 \cos(\theta)} \) defines a hyperbolic path in polar coordinates. As \( \theta \) changes, \( r \) adjusts accordingly to describe different points on the hyperbola. Polar coordinates are particularly elegant for representing a hyperbola because they deal smoothly with infinite extensions and rotations inherent to hyperbolic shapes. Understanding polar coordinates is crucial for appreciating the dynamics and symmetry of conic sections such as hyperbolas.

Hyperbola

A hyperbola is one of the four basic conic sections that can be defined by its geometric properties. Unlike circles and ellipses, hyperbolas expand without bound, often described as having two distinct branches. Hyperbolas arise in various contexts in mathematics and science.
- Hyperbolas are defined mathematically by the condition that the difference in distances from any point on the hyperbola to the two focal points is constant.
- In the polar form, hyperbolas have an eccentricity greater than 1, distinguishing them from other conic sections.
The equation for this exercise \( r = \frac{6}{1 + 3 \cos(\theta)} \) describes a hyperbola because its eccentricity (\( e = 3 \)) is greater than 1. The equation reveals insights into the hyperbola's behavior and orientation. For instance, the \( \cos(\theta) \) term suggests a symmetry around the horizontal axis, with one branch of the hyperbola extending to the right of the pole.
Hyperbolas are significant for various real-world applications, including in satellite trajectories and radio wave propagation. Understanding hyperbolas through their polar equations is fundamental for deeper insights into their properties and behaviors.