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 2; the conic is a hyperbola.

Step 1: Identify Form and Determine Parameters

The given polar equation is \( r = \frac{8}{3 + \cos \theta} \). This is in the form \( r = \frac{ed}{1 + e \cos \theta} \) where \( e \) is the eccentricity and \( d \) is a constant related to the directrix of the conic. Compare to find \( e \) and \( ed = 8 \).

Step 2: Solve for Eccentricity

Since the equation is in the form \( r = \frac{ed}{1 + e \cos \theta}\), we see that the denominator matches \(3 + \cos \theta\) from \(1 + e \cos \theta\). Thus, we equate \(1 + e = 3\) which solves to \(e = 2\).

Step 3: Identify the Conic Type

For a conic in polar coordinates, if \(e = 1\) it is a parabola, \(e < 1\) it is an ellipse, \(e = 1\) it is a parabola, and \(e > 1\) it is a hyperbola. Since \(e = 2 \) which is greater than 1, the conic is a hyperbola.

Step 4: Identify Constant and Relation to Directrix

With \(ed = 8\) and \(e = 2\), we solve for \(d\). So, \(2d = 8\), which gives \(d = 4\). The directrix is one and vertex is on one of the major axes.

Step 5: Sketch the Conic and Label the Vertices

For \(r = \frac{8}{3 + \cos \theta}\), plot the hyperbola with the center at the pole (origin). The conic is horizontal and opens to the right because the \(\cos\) function is involved. Label the vertices (major axis intersections) according to \(8\) and \(\cos\theta = 1\). The conic intersects the \(\theta = 0\) line at the vertices.

Key Concepts

Eccentricity

Eccentricity is a critical concept when studying conic sections like ellipses, parabolas, and hyperbolas. It is a number that describes the shape of a conic section.
For instance, the eccentricity (\(e\)) tells us how much a conic section deviates from being circular.
An important reminder when working with eccentricities:

• For a circle, the eccentricity (\(e\)) is 0, which means no deviation.
• An ellipse has an eccentricity between 0 and 1.
• A parabola has an eccentricity exactly equal to 1.
• Hyperbolas, like the one given in our exercise, have eccentricities greater than 1.

In the given problem, the polar equation is of the form \( r = \frac{ed}{1 + e \cos \theta} \), where \(e=2\). This means our conic is a hyperbola since \(e\) is greater than 1.

Polar Coordinates

Polar coordinates provide a way to represent points in a plane using a combination of radius and angle, as opposed to Cartesian coordinates which use x and y coordinates. This system is particularly useful for understanding conic sections, as their curve can often be represented with simpler equations in polar form.
In the polar system:

• \(r\) represents the radius or the distance from the origin to the point.
• \(\theta\) is the angle from the positive x-axis to the point.

The polar equation given in the exercise is \( r = \frac{8}{3 + \cos \theta} \).
This type of formula enables us to work easily with curves such as ellipses and hyperbolas by manipulating the terms involving \(\cos \theta\) or \(\sin \theta\). Using this representation helps to identify essential elements like focus and directrix efficiently.

Hyperbola

A hyperbola is one of the four conic sections that can be formed by intersecting a plane with a double-napped cone. Hyperbolas are often characterized by their two symmetric, open-ended curves.
In polar coordinates, like the equation in our exercise \( r = \frac{8}{3 + \cos \theta} \), hyperbolas have characteristics that can be easily identified:

• The value of the eccentricity \(e\), which is 2 in this case, confirms that it's a hyperbola because \(e > 1\).
• The graph of the polar equation shows the hyperbola opening horizontally, influenced by the presence of the \(\cos \theta\) term.

Hyperbolas have certain distinct parts including:

• Foci, or points that help define the shape.
• Vertex, the point where the hyperbola turns, located on the major axis.
• Asymptotes, which are lines the curve approaches but never meets.

By understanding these key elements, students can effectively sketch and analyze hyperbolas, just like in the given exercise.