Problem 19 - Precalculus

Question

Identify the conic and sketch its graph. $$r=\frac{2}{2-\cos \theta}$$

Step-by-Step Solution

Answer

The given polar equation represents an ellipse with a semi-latus rectum of 2 units and an eccentricity of 1/2. The ellipse has its major axis along the x-axis with its focus at the origin.

Step 1: Identify the Conic Section

The general formula for a conic in polar coordinates is given as \( r = {l \over (1 + e \cos \theta)} \) where \(e\) is the eccentricity and \(l\) is the semi-latus rectum, which indicates the distance from the focus to the curve. To identify the type of the conic section, we need to compare the given equation \( r = {2 \over (2 - \cos \theta)} \) with the general conic formula. Hence, we can rewrite the equation as \(r = {2 / \over (1 - (1/2) \cos \theta)}\). Comparing this with the general conic formula, we can see that \(e = 1/2\) and \(l = 2\), therefore it is an ellipse because the eccentricity \(e\) is less than 1.

Step 2: Sketch the Graph

Since this is a conic with a horizontal directrix, it opens along the x-axis (the direction of \(\theta\)). Because this is an ellipse, it is symmetrical about the x-axis. Hence, the graph of this polar equation will be an ellipse that has its major axis along the x-axis and its focus at the origin.

Key Concepts

Polar Coordinates

Polar coordinates provide an alternative way of plotting points and equations in the plane. Unlike Cartesian coordinates, which use a grid of x and y values, polar coordinates use a radius and an angle to define a point. This can simplify equations and the visualization of certain figures, like conic sections.

In polar coordinates, each point in the plane is determined by:

• The distance from the origin (radius or "r").
• The angle from the positive x-axis (angle or "\(\theta\)").

A polar equation describes the relationship between \(r\) (the radius) and \(\theta\) (the angle). Conic sections are often described using such equations. For example, an equation of the form \( r = \frac{l}{1 + e \cos \theta} \) is typical for conic sections in polar coordinates. This formula helps identify different conic sections like ellipses, parabolas, and hyperbolas.

Using polar coordinates can make identifying and graphing conic sections straightforward by directly utilizing the parameters that define their shapes, such as eccentricity and the semi-latus rectum.

Eccentricity

Eccentricity is a key concept when dealing with conic sections, and it helps us classify them. It is a number that describes how much a conic section deviates from being a perfect circle. The value of eccentricity, usually represented by \(e\), distinguishes one conic from another.

Here’s how it works for each type of conic section:

• An ellipse has an eccentricity less than 1 (\(0 < e < 1\)). The smaller the eccentricity, the closer the ellipse is to being a circle.
• A parabola has an eccentricity equal to 1 (\(e = 1\)).
• A hyperbola has an eccentricity greater than 1 (\(e > 1\)).

In the provided polar equation \(r = \frac{2}{2 - \cos \theta}\), we compared it with the general polar equation \(r = \frac{l}{1 + e \cos \theta}\) and found \(e = \frac{1}{2}\). Since \(e < 1\), the conic is an ellipse. Understanding the role of eccentricity is crucial for identifying the type and shape of a conic section.

Ellipse

An ellipse is one of the fundamental shapes studied in geometry. It looks like an elongated circle or an oval and is characterized by its two focal points. All points on the ellipse have the property that the sum of their distances to the two foci is constant.

When described in polar coordinates, an ellipse equation takes a specific form: \( r = \frac{l}{1 + e \cos \theta} \). In this form, \(e\) represents eccentricity, and \(l\) is the semi-latus rectum, governing the size and extent of the ellipse. The lesser the eccentricity \(e\), the more the ellipse resembles a circle.

Ellipses have several important features:

• Major and minor axes: The longest and shortest diameters of the ellipse.
• Foci: The two points inside the ellipse used in its definition.
• Directrix: A reference line outside the ellipse used in its equation.

In the problem we solved, with \(e = \frac{1}{2}\), the conic section is an ellipse. This ellipse has its major axis along the x-axis and its focus at the origin, giving it a unique orientation and symmetry in space.