Problem 30 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

(a) The eccentricity is 1, indicating a parabola. (b) Vertex is at \( r = \frac{4}{3} \) for \( \theta = 0 \).

Step 1: Identify the conic equation form

The given equation is of the form \( r = \frac{ed}{1 + e \cos \theta} \) where \( e \) is the eccentricity. We can rewrite the given equation \( r = \frac{8}{3 + 3 \cos \theta} \) as \( r = \frac{\frac{8}{3}}{1 + \cos \theta} \). Hence, \( ed = \frac{8}{3} \) and \( e = 1 \).

Step 2: Determine the eccentricity

For the rewritten equation \( r = \frac{\frac{8}{3}}{1 + \cos \theta} \), compare it with \( r = \frac{ed}{1 + e \cos \theta} \) to find that \( e = 1 \).

Step 3: Identify the type of conic

Since the eccentricity \( e = 1 \), this implies the conic is a parabola. Eccentricity of 1 characterizes a parabola.

Step 4: Determine vertices of the conic

For a parabola, the vertex is the point closest to the focus. In polar form, this corresponds to the minimum distance \( r \) when \( \cos \theta = 1 \), thus \( r = \frac{8}{6} = \frac{4}{3} \).

Step 5: Sketch the conic

To sketch the parabola, note the vertex at \( \theta = 0 \) and \( r = \frac{4}{3} \), which represents the closest point to the focus of the parabola.

Key Concepts

Eccentricity

Eccentricity is a key concept when studying conic sections. It helps us to determine the shape of the conic: circle, ellipse, parabola, or hyperbola. Mathematically, eccentricity is denoted by \( e \). It quantifies how "stretched" a conic section is. Here’s how eccentricity works for different conics:

• If \( e = 0 \), the conic is a circle. Circles are perfectly round and have zero eccentricity since they have equal distance from the center to any point on the circumference.
• If \( 0 < e < 1 \), the conic is an ellipse. Ellipses resemble stretched circles.
• If \( e = 1 \), the conic is a parabola. Parabolas represent the threshold between open and closed curves.
• If \( e > 1 \), the conic is a hyperbola. Hyperbolas consist of two disconnected curves that open away from one another.

In the equation \( r = \frac{8}{3 + 3 \cos \theta} \), we determined that the eccentricity \( e = 1 \). Thus, the conic section is a parabola.

Understanding eccentricity aids in predicting and analyzing the behavior of conic sections with various applications in physics, engineering, and astronomy.

Parabola

A parabola is one of the fundamental shapes in conic sections, defined as the locus of points equidistant from a fixed point called the focus and a line known as the directrix. In polar coordinates, parabolas can be represented using the equation:

• \( r = \frac{ed}{1 + e \cos \theta} \) or \( r = \frac{ed}{1 - e \cos \theta} \)

For a parabola, the eccentricity \( e \) is always 1.

In the given problem, the equation \( r = \frac{8}{3 + 3 \cos \theta} \) was expressed in a form that clearly shows the parabola's characteristics. The vertex of a parabola in polar form is the nearest point to the focus. Calculating this for our equation, we find \( r = \frac{4}{3} \) when \( \theta = 0 \).

Parabolas have many practical uses, including the paths of planetary bodies, the design of satellite dishes, and vehicle headlights, as the symmetrical property of parabolas allows for efficient focus and reflection of paths.

Polar Coordinates

Polar coordinates are a system for representing points in a plane using a distance and an angle. Different from the Cartesian \((x, y)\) system, in polar coordinates, a point is defined by \((r, \theta)\), where:

• \( r \) is the distance from the origin.
• \( \theta \) is the angle with respect to the positive x-axis.

This system is particularly useful for problems involving circular or rotational symmetry. The equation given in the exercise, \( r = \frac{8}{3 + 3 \cos \theta} \), is a polar equation, which suits analyzing properties of conic sections like parabolas using angles and radial distances.

Polar coordinates simplify the process of sketching conics centered at the origin especially when rotations or angles are involved. In our problem, the polar form highlights \( r \), the radial distance, is minimized when \( \theta = 0 \). This gives us a very convenient way to determine the vertex of the parabola in polar coordinates.

Understanding polar coordinates can be beneficial in various fields such as navigation, mechanics, and electromagnetism, as it simplifies calculations and provides intuitive insight into rotational dynamics.