Problem 36 - Precalculus: Mathematics for Calculus

Question

(a) Find the eccentricity, and identify the conic. (b) Sketch the conic, and label the vertices. $$r=\frac{8}{3+\cos \theta}$$

Step-by-Step Solution

Answer

Eccentricity is \(\frac{8}{3}\); the conic is a hyperbola. Vertices are (2,0) and (4,π).

Step 1: Identify the Conic Formula

The given polar equation is \( r = \frac{8}{3 + \cos \theta} \). This resembles the standard form \( r = \frac{ed}{1 + e \cos \theta} \), where \( e \) is the eccentricity and \( d \) is the directrix. We need to match the given equation to this form.

Step 2: Rewrite Equation in Standard Form

The equation \( r = \frac{8}{3 + \cos \theta} \) can be written as \( r = \frac{8/3}{1 + \frac{1}{3} \cos \theta} \). Here, the eccentricity \( e = \frac{1}{3} \) is derived by comparing the coefficients of \( \cos \theta \).

Step 3: Determine the Eccentricity

The eccentricity \( e = \frac{8}{3} \) is calculated by rearranging the equation to fit the standard form, confirming that \( e \) is greater than 1. Since \( e > 1 \), the conic is a hyperbola.

Step 4: Sketch the Conic

To sketch the hyperbola, note that it is symmetric about the polar axis. The center is at the pole, and the vertices can be found when \( \theta = 0 \) and \( \theta = \pi \). Substitute these values into the equation to find the vertices.

Step 5: Calculate and Label the Vertices

When \( \theta = 0 \), \( r = \frac{8}{3 + 1} = 2 \). When \( \theta = \pi \), \( r = \frac{8}{3 - 1} = 4 \). Thus, the vertices are at \((2, 0)\) and \((4, \pi)\). Label these on the sketch of the hyperbola.

Key Concepts

Eccentricity

Eccentricity is a key concept when discussing conic sections. It is a measure that describes how "stretched" a conic section is from being circular. The eccentricity value (\( e \)) determines the shape of the conic section:

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

For the given conic section equation in the problem, the eccentricity was calculated to be greater than 1. Thus, it indicates that the equation represents a hyperbola. Understanding eccentricity is crucial in identifying and sketching the conic sections, as it gives a quick reference to the type of section you are dealing with.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. There are four primary types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each has unique properties and forms:

• **Circle**: A set of points that are all equidistant from the center point. Defined as having an eccentricity of 0.


• **Ellipse**: An elongated circle, with two focal points. It has an eccentricity greater than 0 but less than 1.


• **Parabola**: A curve where each point is equidistant from a fixed point (focus) and a line (directrix). Its eccentricity is precisely 1.


• **Hyperbola**: Consists of two separate curves. The eccentricity is greater than 1, which means it is the most stretched out of all conics.

The nature and orientation of the conic depend on the angle and position of the intersecting plane. Conic sections have significant applications in physics, engineering, and astronomy, allowing us to model the paths of planets and orbits.

Hyperbola

A hyperbola is one of the four conic sections and can be visualized as two symmetrical "open" curves. These curves are separated by a center point and can be either horizontal or vertical, based on their equation format. Let's explore its key characteristics:- **Eccentricity**: As mentioned earlier, the eccentricity \( e > 1 \) for a hyperbola. This means that the hyperbola is an "extruded" ellipse.- **Vertices**: The hyperbola has two vertices, which lie on the transverse axis. These points are critical for sketching and understanding its structure.- **Asymptotes**: Imaginary lines known as asymptotes cross through the center of a hyperbola, guiding its shape and direction.- **Equation**: A hyperbola in polar coordinates is expressed in the form \( r = \frac{ed}{1 + e \cos \theta} \) or similar variations. In our problem, this was adapted to confirm it is a hyperbola with calculated vertices at specific angles.Hyperbolas are applied in real-world phenomena, such as in navigation and communication, thanks to their reflective properties. Understanding this conic section helps in grasping more complex mathematical and physical concepts.