Problem 16 - Algebra and Trigonometry

Question

15–22 (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

The conic is a parabola with eccentricity 1. Its vertex is at (1, 0).

Step 1: Identify the Type of Conic

The general form of a conic in polar coordinates is given by \( r = \frac{ed}{1 + e \cos \theta} \) or \( r = \frac{ed}{1 + e \sin \theta} \). In this exercise, the equation is \( r = \frac{8}{3 + 3 \cos \theta} \). This matches the form \( r = \frac{ed}{1 + e \cos \theta} \) with \( e = \frac{3}{3} = 1 \) and \( ed = 8 \).

Step 2: Find the Eccentricity

From the form of the equation, we have \( e = 1 \). The eccentricity \( e \) determines the type of conic:- If \( e < 1 \), it is an ellipse.- If \( e = 1 \), it is a parabola.- If \( e > 1 \), it is a hyperbola.Thus, since \( e = 1 \), the conic is a parabola.

Step 3: Identify the Conic

Using the eccentricity found in Step 2, we identify that the conic represented by this polar equation is a parabola, because \( e = 1 \).

Step 4: Find and Label the Vertices

For a parabola, the vertex is at the point closest to the directrix. In polar coordinates, when \( \theta = 0 \), \( r = \frac{8}{3 + 3} = 1 \). Therefore, the vertex of the parabola is at \( (1, 0) \) on the polar grid.

Step 5: Sketch the Conic

To sketch the parabola in polar coordinates, plot the point \((1, 0)\), which represents the vertex. Since it is a parabola opening to the right (because it is symmetric about the \(\theta = 0\) axis), draw a smooth curve beginning at the vertex and extending outward, maintaining the shape of a parabola.

Key Concepts

Polar Coordinates

Polar coordinates provide a unique perspective in geometry where points on a plane are determined not by traditional x and y axes, but by an angle and a distance.

• The angle, \(\theta\), is measured from a fixed direction. Typically, this is taken from the positive x-axis.
• The radial distance, \(r\), is the distance from a fixed point, known as the pole (similar to the origin in Cartesian coordinates).

These coordinates are especially useful in dealing with problems involving symmetry about a point such as the origin. Notably, they are effective when working with objects like circles and spirals, as the equations often become simpler and more intuitive.
For example, in our problem, the equation \(r=\frac{8}{3+3\cos\theta}\) relates these two variables, illustrating how the radial distance changes with the angle, forming a parabola. To convert back to Cartesian coordinates or to explore the equation further, substitutions using \(x = r \cos \theta\) and \(y = r \sin \theta\) can be applied. This allows for cross-understanding different geometrical representations.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone. These include ellipses, parabolas, and hyperbolas, each defined by their unique eccentricity.

• Ellipses have an eccentricity (\(e\)) of less than 1.
• Parabolas have an eccentricity of exactly 1.
• Hyperbolas have an eccentricity greater than 1.

In our exercise, the equation given has the form \(r = \frac{ed}{1 + e \cos \theta}\), which helps determine the shape based on the parameter \(e\). Finding \(e = 1\) means the conic section is a parabola.
The beauty of conic sections lies in their geometric properties and applications. Parabolas, for example, have practical applications in physics and engineering, such as in the design of satellite dishes and headlight reflectors, where their reflective properties are advantageous.

Parabola

A parabola is a significant type of conic section known for its distinct U-shaped curve. This shape is defined using an eccentricity \(e = 1\). Its equation in polar coordinates generally appears as \(r = \frac{ed}{1 + e \cos \theta}\).
Key properties of a parabola include:

• The vertex, which is the highest or lowest point on the graph, depending on its orientation. This can be found where \(\theta = 0\), and \(r\) assumes its minimum value, as seen in the provided problem.
• The parabola's symmetry around its axis, which, in polar problems, helps with sketching and understanding the curve's opening direction (rightward in our problem).

The vertex location in this exercise, \((1, 0)\), indicates the nearest point of the parabola to the directrix, giving a basis for sketching and analyzing its path effectively. Understanding parabolas in polar coordinates can enhance comprehension of their unique geometric forms, crucial for studies involving orbital paths or any scenario that benefits from this precise mathematical representation.