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 \( \frac{2d}{e^2} \).

Step 1: Understand the Geometric Context

The given equation \( r = \frac{ed}{1 + e \cos(\theta - \theta_0)} \) represents a conic section in polar coordinates. Here, \(e\) is the eccentricity, and \(d\) is the focal distance. The latus rectum is the line perpendicular to the major axis through a focus of the conic.

Step 2: Recall the Formula for Latus Rectum

The length of the latus rectum \(L\) for a conic section is given by the formula \( L = \frac{2b^2}{a} \), where \( a \) is the semi-major axis and \( b \) is the semi-minor axis. However, we can express it directly in terms of \( e \) and \( d \) specifically for conics in polar form.

Step 3: Derive Latus Rectum Formula for General Conics

For a conic section described in polar form, the length of the latus rectum is given by \( L = \frac{2d}{e^2} \). This expression arises from geometric properties of the ellipse and hyperbola. For a parabola, it reduces to \( 2d \) as \( e=1 \).

Step 4: Apply the Formula for Given Conic

Using the derived formula in Step 3, substitute \( d \) and \( e \) into \( L = \frac{2d}{e^2} \). Thus, the length of the latus rectum in terms of \( e \) and \( d \) for the given conic is \( \frac{2d}{e^2} \).

Key Concepts

Latus Rectum

The latus rectum is a fundamental concept in the study of conic sections. It is a line segment that passes through a focus of the conic and is perpendicular to the major axis of the conic, stretching to the opposite sides of the focus. Think of it as a line that splits the conic section into symmetrical halves directly at the focal point. The length of the latus rectum is important as it helps to understand the geometry and shape of various conics.To calculate the latus rectum for conics expressed in polar coordinates, the formula varies depending on the type of conic:

• Ellipses and Hyperbolas: For these conics, the length of the latus rectum, denoted as \(L\), can be determined using \( L = \frac{2d}{e^2} \).
• Parabolas: If the conic is a parabola (where the eccentricity \( e = 1 \)), the formula simplifies to \( L = 2d \).

This highlights that despite the complexity of conics, the latus rectum provides a simple yet powerful way to gauge their dimensions.

Polar Coordinates

Polar coordinates provide a different way of describing the locations of points in a plane. Unlike Cartesian coordinates, which use \(x\) and \(y\) coordinates, polar coordinates use a distance from the origin, \(r\), and an angle, \(\theta\), measured from the positive x-axis. This coordinate system is particularly useful for dealing with curves like circles and various conic sections where symmetry can be easily exploited.When we express conic sections in terms of polar coordinates, the general form is:\[ r = \frac{ed}{1 + e\cos(\theta-\theta_0)}\]Here, \(e\) represents the eccentricity, and \(d\) is the focal distance from the origin to the conic's focus. The angle \(\theta_0\) helps to shift the conic in the plane according to its orientation. Polar coordinates simplify the study of conic sections by focusing on symmetry and centrality, providing tangible geometric insights.

Eccentricity

Eccentricity is a crucial parameter that characterizes conic sections, indicating how "stretched" or "squashed" a conic section is relative to a circle. The value of eccentricity, denoted as \(e\), helps distinguish between different types of conics:

• Circle: When \(e = 0\), the conic is a perfect circle. There is no elongation, and the focus coincides directly with the center.
• Ellipse: For \(0 < e < 1\), the conic is an ellipse. As \(e\) increases towards 1, the ellipse becomes more elongated.
• Parabola: When \(e = 1\), we have a parabola. It represents a transitional form where the conic is open-ended.
• Hyperbola: If \(e > 1\), the conic is a hyperbola. As \(e\) grows, the two separate curves of the hyperbola become more divergent.

Understanding eccentricity provides significant insights into the geometry and nature of a conic section, offering a way to classify them readily based on their geometric properties.