Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the length of the latus rectum for the general conic \(r=e d /\left[1+e \cos \left(\theta-\theta_{0}\right)\right]\) in terms of \(e\) and \(d\).

Step-by-Step Solution

Answer

The length of the latus rectum is \(\frac{2d}{e}\).

Step 1: Recall the Definition of the Latus Rectum

The latus rectum of a conic section is a line segment perpendicular to the major axis that passes through one of the foci and has its endpoints on the conic.

Step 2: Identify the Elements in the Equation

Analyze the given polar equation of the conic: \( r = \frac{ed}{1 + e \cos(\theta - \theta_0)} \). Here, \(e\) is the eccentricity, and \(d\) is the distance parameter related to the directrix.

Step 3: Relate it to the Conic Sections

This equation represents a conic with eccentricity \(e\). For an ellipse (\(e < 1\)), parabola (\(e = 1\)), or hyperbola (\(e > 1\)), the latus rectum can be determined in terms of \(e\) and \(d\).

Step 4: Calculate the Latus Rectum Length

For any conic, the length of the latus rectum \(L\) is given by the formula \(L = \frac{2b^2}{a}\) where \(b^2 = a^2(1 - e^2)\). In polar form, the eccentricity \(e\) helps us find that \(L = \frac{2d}{e}\).

Step 5: Simplify the Expression

Using the relation from Step 4, write the length of the latus rectum solely in terms of \(e\) and \(d\). Therefore, \(L = \frac{2d}{e}\).

Key Concepts

Latus Rectum

The latus rectum is a term from geometry that describes a specific line segment associated with conic sections, such as ellipses, parabolas, and hyperbolas. It is a line segment that is perpendicular to the major axis of the conic section and passes through one of its foci.
The length of the latus rectum is crucial because it gives us helpful information about the size and shape of conics. For example, in a parabola, the latus rectum length is critical in determining how "open" or "narrow" the parabola is.

• For a parabola, the latus rectum is directly proportional to the distance to the focus.
• For ellipses and hyperbolas, it relates to both the semi-major axis and the eccentricity.

Knowing how to find the latus rectum in polar coordinates involves understanding parameters like eccentricity (\(e\)) and the directrix parameter (\(d\)). When you have a conic in polar form, such as \(r = \frac{ed}{1+e \cos(\theta-\theta_0)}\), the length of the latus rectum \(L\) is given by the expression \(L = \frac{2d}{e}\).
This equation is powerful because it expresses the latus rectum's length solely using the eccentricity and the directrix parameter, facilitating easy calculations across different types of conics.

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where each point is determined by a distance and an angle. This system is particularly useful in situations where relationships are periodic or radial in nature, such as in the analysis of conics. A point in polar coordinates is defined by:

• \(r\) = the radial distance from the origin to the point.
• \(\theta\) = the angle measured from the positive x-axis.

Understanding conics in polar coordinates allows us to analyze their properties and behavior from a different perspective than the usual Cartesian system. The polar form of a conic section equation for a conic with focus at the pole, such as \(r = \frac{ed}{1+e \cos(\theta-\theta_0)}\), helps us visualize the shape by changing both the angle and the distance.
This form is particularly beneficial when dealing with problems like satellite orbits, where the path is naturally circular or elliptical. It also simplifies the process of identifying the type of conic by examining the parameter \(e\), known as eccentricity.

Eccentricity

Eccentricity is a fundamental concept in the study of conics. It is a numerical parameter that defines the shape and type of a conic section. Denoted as \(e\), eccentricity determines how "stretched" a conic is.
Here's how the values of \(e\) define different conics:

• If \(e = 0\), the conic is a circle. It is perfectly round.
• If \(0 < e < 1\), it is an ellipse, which is an elongated circle.
• If \(e = 1\), the conic is a parabola, open-ended and U-shaped.
• If \(e > 1\), it is a hyperbola, consisting of two opposing open curves.

In the polar equation \(r = \frac{ed}{1+e \cos(\theta-\theta_0)}\), eccentricity not only helps determine the type of conic but is also crucial in calculating other properties such as the latus rectum. For instance, the length of the latus rectum, \(L\), is expressed as \(L = \frac{2d}{e}\), highlighting the link between eccentricity and other geometric traits in a conic.