Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(r_{1}\) and \(r_{2}\) be the minimum and maximum distances (perihelion and aphelion, respectively) of the ellipse \(r=\) \(e d /\left[1+e \cos \left(\theta-\theta_{0}\right)\right]\) from a focus. Show that (a) \(r_{1}=e d /(1+e), r_{2}=e d /(1-e)\), (b) major diameter \(=2 e d /\left(1-e^{2}\right)\) and minor diameter \(=\) \(2 e d / \sqrt{1-e^{2}}\)

Step-by-Step Solution

Answer

\(r_1 = \frac{ed}{1-e}\), \(r_2 = \frac{ed}{1+e}\), Major diameter is \(\frac{2ed}{1-e^2}\) and minor diameter is \(\frac{2ed}{\sqrt{1-e^2}}\).

Step 1: Identify Given Equation

The problem provides us the equation of the ellipse, given in polar coordinates as \( r = \frac{ed}{1+e \cos(\theta-\theta_0)} \). We are tasked with finding expressions for the perihelion and aphelion distances, and for the major and minor diameters.

Step 2: Find Perihelion Distance (r1)

For the perihelion distance, \( r_1 \), the minor distance is where \( \theta = \theta_0 + \pi \), since this maximizes \( \cos(\theta - \theta_0) \) to its positive extreme, thus \( \cos(\theta - \theta_0) = -1 \). Solving gives: \[ r_1 = \frac{ed}{1+e(-1)} = \frac{ed}{1-e} \].

Step 3: Find Aphelion Distance (r2)

For the aphelion distance, \( r_2 \), the major distance is when \( \theta = \theta_0 \), maximizing \( \cos(\theta - \theta_0) \) to its negative extreme, thus \( \cos(\theta - \theta_0) = 1 \). Solving gives: \[ r_2 = \frac{ed}{1+e(1)} = \frac{ed}{1+e} \].

Step 4: Calculate Major Diameter

The major diameter, which is the major axis of the ellipse, is simply the sum of \( r_1 \) and \( r_2 \). Thus: \[ \text{Major diameter} = r_1 + r_2 = \frac{ed}{1+e} + \frac{ed}{1-e} = \frac{2ed}{1-e^2} \].

Step 5: Derive Minor Diameter

Using the relation for components of an ellipse in polar coordinates, the minor diameter can be found using: \[ \text{Minor diameter} = 2 \times \frac{ed}{\sqrt{1-e^2}} \].This relation comes from the rearrangement and simplification based on the ellipse's geometry and the identity for ellipses.

Key Concepts

Perihelion and Aphelion Distances

In the context of an ellipse, the terms perihelion and aphelion distances are fundamentally crucial. They refer to the closest and farthest points of the ellipse from one of its foci. Knowing these distances is vital as they simplify understanding how elongated the ellipse is.
To locate these distances mathematically, we use the ellipse equation in polar coordinates. When you have such an equation as \( r = \frac{ed}{1+e \cos(\theta-\theta_0)} \):

• **Perihelion Distance (\(r_1\))** occurs when the value of \( \cos(\theta-\theta_0) \) minimizes to its lowest point, effectively translating to \( \cos(\theta-\theta_0) = -1 \).
• **Aphelion Distance (\(r_2\))** is found when \( \cos(\theta-\theta_0) \) maximizes to its highest point, specifically turning into \( \cos(\theta-\theta_0) = 1 \).

Ultimately, these principles are used to derive the expressions for \( r_1 = \frac{ed}{1-e} \) and \( r_2 = \frac{ed}{1+e} \), offering an easy means to determine how an ellipse stretches around a focus.

Major and Minor Diameters

Major and minor diameters are essential for defining the shape size and characteristics of an ellipse.

The **major diameter** is the longest diameter of the ellipse. It stretches through the center and both foci, from one side of the ellipse to the other. To compute the major diameter of an ellipse given in polar coordinates, you add together the perihelion and aphelion distances of the ellipse.

• The formula for the major diameter becomes \( \text{Major diameter} = \frac{2ed}{1-e^2} \).

The **minor diameter** is perpendicular to the major diameter, spanning the shortest path through the center of the ellipse.

• To calculate the minor diameter, apply the relation \( \text{Minor diameter} = 2 \times \frac{ed}{\sqrt{1-e^2}} \)

Understanding these diameters helps in comprehending the overall dimensions and eccentricity of an ellipse.

Polar Coordinates in Ellipses

Polar coordinates offer a unique way to describe the position and shape characteristics of an ellipse, especially when one focus is at the pole. This results in equations that are particularly useful for various applications, such as astronomy.

The essence of using polar coordinates is reflected in the equation \( r = \frac{ed}{1+e \cos(\theta-\theta_0)} \), where:\

• \( r \) represents the radius vector, the distance from the pole to a point on the ellipse.
• \( e \) is the eccentricity of the ellipse, a measure of how much the shape deviates from being circular.
• \( \theta \) is the polar angle, and \( \theta_0 \) is the angle where this form of the equation is minimized or maximized.

Expressing ellipses in polar form with these elements allows for easy calculation of distances like perihelion and aphelion, as well as insights into the ellipse's geometric properties.