Problem 15 - Calculus Early Transcendentals

Question

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$r=\frac{3}{4-8 \cos \theta}$$

Step-by-Step Solution

Answer

Eccentricity is 8, it's a hyperbola, directrix is \( x = \frac{3}{8} \).

Step 1: Identify the Conic Type

The polar equation is given by \( r = \frac{ed}{1 - e\cos\theta} \). By comparing, we have \( e = 8 \) and \( ed = 3 \). This represents a conic in polar form.

Step 2: Find the Eccentricity

The given equation is already in the form where \( e = 8 \). Since \( e > 1 \), this is the eccentricity of the hyperbola.

Step 3: Verify Conic Type

Since \( e > 1 \), the conic is a hyperbola. The value supports the identification step.

Step 4: Calculate the Directrix

Using \( ed = 3 \) from our equation, solve for \( d \): \( d = \frac{3}{8} \). Since the equation is \( r = \frac{ed}{1 - e\cos\theta} \), the directrix corresponding to a horizontal hyperbola is \( x = \frac{3}{8} \).

Step 5: Sketch the Conic

Plot the equation \( r = \frac{3}{4 - 8\cos\theta} \) for values of \( \theta \) from \( 0 \) to \( 2\pi \). The conic will display two branches opening horizontally, indicating a hyperbola.

Key Concepts

Eccentricity

Eccentricity is a measure of how much a conic section deviates from being a circle. For any conic section, the eccentricity (\( e \) ) determines its shape:

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

In the given problem, we identify\( e = 8 \) from the equation \( r=\frac{3}{4-8 \cos \theta} \) . This means our conic is a hyperbola because,simply put, \( 8 > 1 \).Breaking up text helps comprehension.
It also leads to better readability.

Hyperbola

A hyperbola is a type of conic section that forms two separate curves, called branches, that mirror each other. They open in opposite directions and are defined by their eccentricity, which is greater than one.

• Compared to an ellipse, hyperbolas are more stretched and continue indefinitely in opposite directions.
• The hyperbola has two asymptotes that cross at its center. These lines are a guide for the directions of its arms.
• Each branch comes closest to these asymptotes but never actually meets them.

In the case of the problem, the equation \( r=\frac{3}{4-8 \cos \theta} \) defines a hyperbola in polar coordinates.
This hyperbola opens in the horizontal direction, which is consistent with the cosine function in the denominator.

Polar Coordinates

Polar coordinates offer a unique way to define a location in a plane using a radius and angle, unlike Cartesian coordinates that use x and y axes.

• In polar systems, any point is defined by \( (r, \theta) \) , where \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis.
• This system is especially useful for expressing circular and radial shapes, making it ideal for conic sections.

The conic equation provided \( r=\frac{3}{4-8 \cos \theta} \) effectively uses polar coordinates.
The variable \( \theta \) represents the angle, and \( r \) is the radial distance from a pivotal point, usual for hyperbolas.

Directrix

The directrix of a conic section is a fixed line used to construct and define the curve of the conic. It, along with the eccentricity and focus, defines the path the conic takes.

• For hyperbolas, the directrix helps in shaping the open part of the curves.
• In a conventional setting, the directrix is perpendicular to the axis of symmetry of the hyperbola.

For the equation at hand, \( r=\frac{3}{4-8 \cos \theta} \), we find the directrix using \( ed = 3 \).By solving \( d = \frac{3}{8} \), it shows the directrix as \( x = \frac{3}{8} \).
This line corresponds directly to the way the hyperbola opens, being horizontal in this example.