Problem 25 - Precalculus: Functions and Graphs

Question

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$e=\frac{1}{3}, \quad r=2 \sec \theta$$

Step-by-Step Solution

Answer

The polar equation is \(r = \frac{2}{3 + \cos \theta}\).

Step 1: Identify the provided information

We are given the eccentricity \(e = \frac{1}{3}\) and the equation of the directrix \(r = 2 \sec \theta\). In polar coordinates, \(r = d \sec \theta\) represents a vertical directrix at \(x = d\), where \(d = 2\).

Step 2: Determine the conic type

Since the eccentricity \(e = \frac{1}{3} < 1\), the conic is an ellipse. A conic is determined by its eccentricity: \(e < 1\) for an ellipse, \(e = 1\) for a parabola, and \(e > 1\) for a hyperbola.

Step 3: Use the polar equation for a conic with directrix x = d

The general polar equation of a conic with a directrix \(x = d\) and focus at the pole is given by \[ r = \frac{ed}{1 - e \cos \theta} \]. Here, \(e = \frac{1}{3}\) and \(d = 2\).

Step 4: Substitute values into the polar equation

Substitute the values of \(e\) and \(d\) into the polar equation: \[ r = \frac{\frac{1}{3} \times 2}{1 - \frac{1}{3} \cos \theta} = \frac{\frac{2}{3}}{1 - \frac{1}{3} \cos \theta} \]. Simplify the expression: \[ r = \frac{2}{3 + \cos \theta} \].

Key Concepts

Eccentricity

Eccentricity is a fundamental concept when studying conics. It defines the shape of the conic section by measuring the deviation of the conic from being a perfect circle. In simple terms, it tells us how "stretched" the conic is.

Here's a quick breakdown:

• An eccentricity (\( e \)) of less than 1 indicates an ellipse, which is more circular in shape.
• An eccentricity of exactly 1 corresponds to a parabola, which opens wider but does not "close."
• For an eccentricity greater than 1, the conic is a hyperbola, characterized by its open and spread-out shape.

In our exercise, we were given an eccentricity of \( e = \frac{1}{3} \), which is clearly less than 1. Therefore, the conic section is an ellipse. Remember, the smaller the eccentricity, the more the ellipse resembles a circle.

Ellipse

An ellipse is one of the most common types of conic sections and can be thought of as a squished circle. Its unique property is that the sum of the distances from any point on the ellipse to the two foci is constant. This makes ellipses very special and useful in various scientific fields, including astronomy where the orbits of planets are elliptical.

In polar coordinates, ellipses have specific equations that incorporate their eccentricity and directrix. The polar form of an ellipse with eccentricity \( e < 1 \), and a focus at the origin, is given by:\[ r = \frac{ed}{1 - e \cos \theta}\]Here, \( r \) represents the radius or the distance from the pole (the focus) to any point on the ellipse, and \( \theta \) is the angle from the polar axis.

Understanding this formula allows us to graph and analyze ellipses efficiently.

Directrix

The concept of a directrix is often less familiar to students, but it's crucial for defining conics. A directrix is a line that, together with a focus, helps in describing the conic.

In essence, a conic section can be defined as the set of all points that have a constant ratio known as the eccentricity (\( e \)) to their distances from a fixed point (the focus) and a fixed line (the directrix). For our particular ellipse, the directrix is given by \( r = 2 \sec \theta \), which translates to a vertical line at \( x = 2 \) in Cartesian coordinates.

Understanding the role of the directrix can help better visualize and draw the conic. It is perpendicular to the axis of symmetry of the conic and acts as a guide to maintain the "shape" imposed by the focus.