Question

For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. $$ r=\frac{3}{2-6 \sin \theta} $$

Short answer

The eccentricity is 3, so the conic is a hyperbola.

Step-by-step solution

Identify the Polar Equation Form

The given equation is \( r = \frac{3}{2 - 6 \sin \theta} \). In polar coordinates, the general form of a conic is \( r = \frac{ed}{1 - e \sin \theta} \) or \( r = \frac{ed}{1 - e \cos \theta} \), where \( e \) is the eccentricity, and \( d \) is a constant.

Rewrite the Equation in Standard Form

Compare \( r = \frac{3}{2 - 6 \sin \theta} \) with the standard form \( r = \frac{ed}{1 - e \sin \theta} \). For this equation to match, we can express it as \( r = \frac{3}{\frac{1}{3} (6 - 2 \sin \theta)} = \frac{\frac{3}{3}}{1 - \frac{6}{2} \sin \theta} = \frac{1}{1 - 2 \sin \theta} \times 3 \). Thus, the equation matches when \( ed = 3 \), \( e = 3 \), and \( 2 = 1 \).

Identify Eccentricity

In this expression, the eccentricity \( e \) is clearly \( 3 \) (since the \( e \) term coefficient before \( \sin \theta \) is 6, and 6 divided by the constant 2 gives 3).

Classify the Conic

The type of conic is determined by its eccentricity, \( e \). If \( e < 1 \), it is an ellipse; if \( e = 1 \), it is a parabola; if \( e > 1 \), it is a hyperbola. Since \( e = 3 \) in this case, which is greater than 1, this conic is a hyperbola.

Key concepts

eccentricity

Eccentricity is a crucial concept when studying conic sections, especially in polar coordinates. It helps in understanding the shape of the conic. The eccentricity value, represented by the letter \( e \), tells us a lot about the shape and type of the conic section we are dealing with.
Eccentricity can be interpreted as a measure of how much the conic section deviates from being circular. Based on the value of \( e \):

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

In the given exercise, the eccentricity \( e \) is calculated to be 3. This value indicates that the conic is a hyperbola because it is greater than 1.
Understanding eccentricity helps us classify conics efficaciously from their equations, making it essential when working with polar forms.

hyperbola

A hyperbola is a specific type of conic section, identified by its unique properties and geometry. One of the main characteristics of a hyperbola is that it has two branches that open away from each other. In contrast to ellipses and parabolas, hyperbolas occur when the eccentricity \( e > 1 \).
This geometric figure has certain distinct properties:

• Two focal points, unlike an ellipse, which are used to define the figure.
• The difference in distances from any point on the hyperbola to the two foci is constant.
• As open curves, they extend infinitely in directions that are opposite to each other.

In polar forms, recognizing a hyperbola depends on identifying the eccentricity. With the given equation in the exercise, we find that \( e \) is 3. This confirms its classification as a hyperbola. Knowing a conic is a hyperbola helps in understanding its physical and graphical behavior, which is essential in various applications.

polar coordinates

Polar coordinates provide an alternative system to describe the locations of points on a plane. While the Cartesian coordinate system uses an \((x, y)\) format to denote points, polar coordinates use \((r, \theta)\).
The polar form is particularly advantageous when dealing with problems involving rotational symmetry or periodic functions. The variables in this system are:

• \( r \): It represents the radial distance from the origin to the point.
• \( \theta \): It is the angle made with the positive x-axis.

In polar form, conic sections are often expressed as equations involving \( r \) and \( \theta \). For example, the given equation, \( r = \frac{3}{2 - 6 \sin \theta} \), is a typical polar equation form for a conic section. It includes parameters like eccentricity \( e \), which is essential for understanding its nature. By expressing conics in polar coordinates, it becomes easier to analyze their properties and identify their types effectively.