Problem 29 - Precalculus: Mathematics for Calculus

Question

(a) Find the eccentricity, and identify the conic. (b) Sketch the conic, and label the vertices. $$r=\frac{4}{1+3 \cos \theta}$$

Step-by-Step Solution

Answer

The conic is a hyperbola with eccentricity 3.

Step 1: Identify the Conic Type

The given equation is in the form \( r = \frac{ed}{1 + e\cos\theta} \). This is the polar form of a conic section with eccentricity \( e \). Here, comparing with the given equation \( r = \frac{4}{1+3\cos\theta} \), we can identify that \( e = 3 \). Since \( e > 1 \), the conic is a hyperbola.

Step 2: Calculate the Eccentricity

The eccentricity of a conic in polar coordinates is directly given by the coefficient of \( \cos\theta \) in the denominator. Therefore, for the equation \( r = \frac{4}{1+3\cos\theta} \), we have \( e = 3 \).

Step 3: Sketch the Conic

To sketch the conic, consider the general shape of hyperbolas in polar coordinates. Since the coefficient of \( \cos\theta \) is positive and greater than 1, the conic opens along the horizontal axis in the polar plane. Sketch two branches of a hyperbola facing left and right.

Step 4: Determine and Label the Vertices

For a hyperbola in polar form \( r = \frac{ed}{1+e\cos\theta} \), the vertices occur at \( \theta = 0 \) and \( \theta = \pi \). Calculate the radii for vertices where \( r_1 = \frac{4}{1 + 3 \cdot 1} = 1 \) and \( r_2 = \frac{4}{1 + 3 \cdot (-1)} = -4 \). Place and label these points on the sketch.

Key Concepts

Eccentricity

Eccentricity is an important parameter in the study of conic sections. It helps to determine the shape and nature of the conic. In simple terms, eccentricity measures how much a conic section deviates from being circular.

Here are the key points to know about eccentricity:

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

In the given problem, using the equation \( r = \frac{4}{1 + 3\cos\theta} \), the eccentricity \( e \) can be directly read as 3.

The fact that \( e = 3 \) clearly indicates that it is a hyperbola since \( e > 1 \). Calculating eccentricity is crucial for identifying the type of conic and understanding its features.

Polar Coordinates

Polar coordinates provide a unique way of representing points using two values: a radial distance from the origin and an angle from a reference direction. This system is especially useful in dealing with equations of conic sections because of its simplicity with angles.

The general form of a conic section in polar coordinates is \( r = \frac{ed}{1 + e\cos\theta} \) or \( r = \frac{ed}{1 + e\sin\theta} \).

• \( r \) stands for the radius or distance from the pole (origin) to a point on the conic.
• \( \theta \) represents the angle measured from the positive x-axis or polar axis.
• \( d \) is related to the semi-major axis of the ellipse or hyperbola.

In the current example, by understanding the polar coordinate form, we can easily identify the nature of the conic section.

The connection between the position in a polar system and a specific conic section helps in sketching figures and predicting their general direction and orientation.

Hyperbola

A hyperbola is a fascinating conic section formed when a plane intersects both cones in a double cone setup. Hyperbolas have two separate branches, each mirroring the other.

Key aspects of a hyperbola include:

• An eccentricity greater than 1.
• Two distinct vertices and two focal points.
• Asymptotes that act as boundary lines which the hyperbola approaches but never crosses.

In the specific problem, the hyperbola is represented in polar coordinates as \( r = \frac{4}{1 + 3\cos\theta} \). Therefore, this equation denotes a hyperbola with a horizontal transverse axis since it is in terms of \( \cos\theta \).
The vertices of the hyperbola from the equation are determined by setting \( \theta = 0 \) and \( \theta = \pi \), resulting in radii of \( r = 1 \) and \( r = -4 \). These positions help construct a complete diagram of the hyperbola, giving a visual representation of its orientation and span in the polar plane.