Problem 19 - Precalculus with Limits

Question

In Exercises 15-28, identify the conic and sketch its graph. \(r=\dfrac{2}{2-\cos\ \theta}\)

Step-by-Step Solution

Answer

From the given equation \(r=\dfrac{2}{2-\cos\ \theta}\), it's clear that it fits the definition of a conic with a horizontal directrix, specifically a parabola. Sketching the equation will result in a parabola graph that opens right.

Step 1: Identify the Type of Conic

Looking at the given polar equation \(r=\dfrac{2}{2-\cos\ \theta}\), it can be seen that the equation fits the form of a conic with a horizontal directrix, \(r = \dfrac{ed}{1±e\cos\(\theta\)}\). In this case, \(e=1\) and \(d=1\), which indicates that the given equation represents a conic with eccentricity 1. Therefore, the conic is a parabola.

Step 2: Sketch the Parabola on a Graph

Since it’s a parabola, when plotted, the graph would look like a U shape. The parabola will be opening to the right if \(e>0\) and to the left if \(e<0\). In this case, the parabola should open to the right. The directrix should be a vertical line to the left of the vertex because it's a horizontal directrix.

Step 3: Double Check with Eccentricity

The eccentricity, e, determines the shape of a conic section. e can be 0 (circle), between 0 and 1 (ellipse), 1 (parabola), or greater than 1 (hyperbola). As stated earlier, e is 1, which matches a parabola. Hence, it's right to say the given equation is a parabola.

Key Concepts

Polar Equations

Polar equations are mathematical expressions that describe curves using the polar coordinate system. Unlike the Cartesian coordinate system, which is based on x and y coordinates, polar coordinates describe a point by its distance from a reference point (the pole or origin) and an angle from a reference direction (usually the positive x-axis).

The equation in the exercise, \(r=\dfrac{2}{2-\cos \theta}\), uses the variables \(r\) for the radius and \(\theta\) for the angle to define the curve. In polar equations for conic sections, the typical formula for a conic is \(r = \dfrac{ed}{1 \pm e\cos(\theta)}\) or \(r = \dfrac{ed}{1 \pm e\sin(\theta)}\). Here, \(e\) is the eccentricity, and \(d\) is the distance from the pole to the directrix. Understanding these variables and how they come together allows us to identify types of conics.

Eccentricity

Eccentricity is a key parameter in understanding conic sections. It specifies the degree of deviation of a conic section from being circular. The eccentricity \(e\) is a non-negative number that can determine the specific type of conic:

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

In our exercise, the eccentricity \(e = 1\), which indicates that the conic section is a parabola. This is a crucial step in analyzing the type of graph that will be plotted from a polar equation.

Parabola

A parabola is a U-shaped curve that has some unique properties. It has exactly one axis of symmetry and one vertex, which is the point at which the parabola changes direction. Parabolas can open upwards, downwards, left, or right depending on the orientation of the equation.

In the context of conic sections described using polar coordinates, a parabola occurs when the equation takes the form \(r = \dfrac{ed}{1 - e\cos(\theta)}\) or something similar, and the eccentricity \(e = 1\). The parabola's directrix (a guiding line outside of the parabola that helps in its construction) is crucial in its formation and depends on the other parameters of the equation. The vertex of the parabola is always equidistant from the directrix and the focus.

Graphing Conics

Graphing conics involves plotting equations of conic sections, such as ellipses, parabolas, and hyperbolas. Each conic section has unique graphing rules based on its equation and parameters like the eccentricity and directrix.

For our specific case, knowing that our polar equation represents a parabola, we need to focus on its orientation and location. The vertex of this parabola will be on the horizontal axis, with the curve opening to the right when graphed. This directional opening is due to the nature of its directrix being vertical to the left of the vertex.

It's important to start by identifying the general shape and orientation through the equation before graphing. By analyzing the polar form and computing key characteristics like the vertex and the axis, the conic can be accurately sketched.