Problem 1 - Precalculus: Mathematical for Calculus

Question

\(1-8=\) Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(\frac{2}{3},\) directrix \(x=3\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{6}{3 - 2 \cos \theta} \).

Step 1: Understanding Eccentricity

We know that conics have a characteristic called eccentricity, denoted as \( e \). For ellipses, the eccentricity is less than 1. In this problem, \( e = \frac{2}{3} \), confirming it's an ellipse, as it's less than 1.

Step 2: Understanding the Directrix

The directrix provided is \( x = 3 \). This implies that for the ellipse described in polar coordinates \( r = \frac{ed}{1 - e \cos \theta} \) or \( r = \frac{ed}{1 - e \sin \theta} \), depending on how the directrix relates to the focus.

Step 3: Determine the Formula Type

Since the directrix is a vertical line \( x = 3 \), it indicates that the conic formula involves \( \cos \theta \). Thus, we use the formula \( r = \frac{ed}{1 - e \cos \theta} \).

Step 4: Calculate Constant \( d \)

The constant \( d \) represents the distance from the focus to the directrix. Since the directrix is \( x = 3 \) and the focus is at the origin, \( d = 3 \).

Step 5: Forming the Polar Equation

Substitute the values derived: \( e = \frac{2}{3} \) and \( d = 3 \) into the polar equation. So, the equation becomes: \[ r = \frac{\left(\frac{2}{3}\right)(3)}{1 - \left(\frac{2}{3}\right)\cos \theta} = \frac{2}{1 - \frac{2}{3}\cos \theta}\] To simplify, multiply numerator and denominator by 3: \[ r = \frac{6}{3 - 2 \cos \theta}\] This is the polar equation of the ellipse.

Key Concepts

Eccentricity

Eccentricity is a crucial concept when discussing conic sections, like ellipses, parabolas, and hyperbolas. It helps to define the shape and nature of the conic. Eccentricity, denoted as \( e \), describes how "stretched" the conic is. For an ellipse, the eccentricity is always between 0 and 1. This denotes that an ellipse is a closed curve that appears round but is stretched along either the horizontal or vertical direction. In simpler terms, it looks like an oval.

For example, if \( e = \frac{2}{3} \), which is the case in our problem, the value is clearly less than 1. This confirms that the curve we are dealing with is an ellipse.

• When \( e = 0 \), the conic is a perfect circle.
• As \( e \) approaches 1, the ellipse becomes more elongated.

The exact shape of the ellipse and how stretched it appears is determined by the eccentricity value, making it an essential attribute in classifying and understanding conics.

Ellipse

An ellipse can be imagined as a squashed circle where all points on its outline are equidistant from two fixed points known as the foci. It is one of the four basic types of conic sections, the others being the circle, parabola, and hyperbola.

The general characteristics of ellipses depend on their eccentricity, as mentioned in the previous section, but they also relate to the orientation and position of the ellipse. In our example, the polar form is used to express the ellipse with its focus at the origin.

The polar equation provided is:

• \[ r = \frac{6}{3 - 2 \cos \theta} \]

This formula shows how distance \( r \) changes based on the angle \( \theta \), making it easier to analyze ellipses with a focus at the origin.
Ellipses are very useful in real-world applications ranging from planetary orbits to architectural designs, providing a smooth and symmetric shape for various purposes.

Directrix

The concept of a directrix provides a unique way to define conics, including ellipses. A directrix is a fixed line used to describe a conic section. Together with a focus point, it helps in maintaining a constant ratio, defined by eccentricity, for any point on the conic.
In our problem, the directrix is given as \( x = 3 \). This vertical line serves as a reference for positioning the ellipse relative to the origin. This affects the equation used to express the conic in polar coordinates.
When a conic's directrix is aligned vertically, as with the line \( x = 3 \):

• The polar form that relates to the directrix is \( r = \frac{ed}{1 - e \cos \theta} \).

In this case, \( d \) is the distance from the origin to the line \( x = 3 \), measured perpendicularly. For us, \( d = 3 \).
Knowing the directrix allows for the accurate placement and definition of the ellipse, ensuring it maintains the necessary geometric properties dictated by its eccentricity.