Problem 287 - Calculus Volume 3

Question

Find a polar equation of the conic with focus at the origin and eccentricity and directrix as given. $$ \text { Directrix: } \mathrm{y}=2 ; e=2 $$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{4}{1 - 2 \sin \theta} \).

Step 1: Identify the Key Elements

For this problem, we are given the directrix \( y = 2 \) and the eccentricity \( e = 2 \). The focus is at the origin. Since the directrix is horizontal, the conic section will be symmetric about the y-axis and we will use \( r = \frac{ed}{1 - e \sin \theta} \) to formulate the polar equation.

Step 2: Define the Variables

In this case, the directrix is \( y = 2 \), providing \( d = 2 \) since it is the distance from the directrix to the origin along the vertical line. The eccentricity is given as \( e = 2 \). In polar coordinates, the relevant trigonometric function for a horizontal directrix is \( \sin \theta \).

Step 3: Substitute into Polar Equation

Using the formula \( r = \frac{ed}{1 - e \sin \theta} \), substitute \( e = 2 \) and \( d = 2 \): \[ r = \frac{2 \times 2}{1 - 2 \sin \theta} \] This simplifies to: \[ r = \frac{4}{1 - 2 \sin \theta} \].

Step 4: Analyze the Conic Type

Since \( e > 1 \), this conic is a hyperbola. The equation \( r = \frac{4}{1 - 2 \sin \theta} \) represents a conic section with the focus at the origin and a horizontal directrix, confirming the conic is a hyperbola.

Key Concepts

Eccentricity

Eccentricity is a vital parameter that helps define the shape and type of a conic section. It is denoted by the letter "e" and is a measure of how much a conic section deviates from being circular.

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

For the given problem, the eccentricity is \( e = 2 \). Since \( e > 1 \), it indicates we are dealing with a hyperbola. Understanding the role of eccentricity is crucial because it tells us the conic’s openness or narrowness in shape. When eccentricity increases beyond 1, the hyperbola opens wider.

Directrix in Polar Coordinates

In Cartesian coordinates, a directrix is a fixed line used to define a conic section. However, directrices can also translate into polar coordinates, providing a valuable reference for determining the conic's equation. For a conic section with a focus at the origin, the position and orientation of the directrix help define the equation in polar form.In this exercise, the directrix is given as a horizontal line \( y = 2 \). Knowing this helps us choose the appropriate trigonometric function in the polar equation: - For a horizontal directrix, we use \( \sin \theta \).- The distance, \( d \), from the origin to the directrix is 2.The choice of using \( \sin \theta \) versus \( \cos \theta \) depends entirely on whether the directrix is horizontal or vertical.

Hyperbolas in Polar Form

A hyperbola in polar coordinates can be expressed with a specific formula that incorporates both the eccentricity and the directrix:\[ r = \frac{ed}{1 - e \sin \theta} \]For this equation:- \( r \) is the radius vector.- \( d \) is the length of the directrix from the pole.- \( \theta \) is the angle.From the provided exercise, after placing \( e = 2 \) and \( d = 2 \) into the equation, we derive:\[ r = \frac{4}{1 - 2 \sin \theta} \]This simplifies to express the hyperbola’s nature in polar coordinates. The condition \( e > 1 \) confirms the conical section is indeed a hyperbola. As \( \theta \) varies, \( r \) describes the points on the hyperbola, capturing how the conic section navigates through space with respect to its focus and directrix.