Problem 94 - Calculus Early Transcendentals

Question

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The length of the latus rectum of a hyperbola centered at the origin is \(2 b^{2} / a=2 b \sqrt{e^{2}-1}\)

Step-by-Step Solution

Answer

Answer: The length of the latus rectum of a hyperbola is given by \(l = 2\sqrt{a^2(e^2 - 1)}\).

Step 1: Write the standard equation of a hyperbola centered at the origin.

The standard equation of a hyperbola centered at the origin and opens horizontally is given by \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Step 2: Identify the relationship between the semi-major axis \(a\), focus \(f\), and eccentricity \(e\) of a hyperbola.

For a hyperbola, the relationship between the semi-major axis, focus, and eccentricity is \(f = a * e\). The focus lies on the x-axis, so the coordinates of the two foci are (\(\pm f, 0\)).

Step 3: Derive the relationship between \(a\), \(b\), and \(e\) for a hyperbola.

Using the foci and the vertices of a hyperbola, we can derive the relationship between \(a, b, e\) as follows: \(e = \frac{f}{a} = \frac{\sqrt{a^2 + b^2}}{a}\), then \(b^2 = a^2(e^2 - 1)\).

Step 4: Find the coordinates of the vertices located on the latus rectum.

The latus rectum is the focal chord perpendicular to the major axis of the hyperbola. We are considering a horizontal hyperbola, so the latus rectum passes through both foci and is parallel to the y-axis. As the foci lie on the x-axis, their coordinates are (\(\pm f, 0\)). The latus rectum shares both foci, implying the end points of the latus rectum and vertices of the hyperbola are on the same horizontal with y-coordinate 0. Let's denote the coordinates of these two points to be (\(f + b\), 0) and (\(f - b\), 0).

Step 5: Compute the length of the latus rectum.

The length of the latus rectum is the distance between the two end points found in Step 4. By applying the distance formula, we have \(l = |(f+b) - (f-b)| = 2b\). Now, we need to express \(b\) in terms of \(a\) and \(e\). From Step 3, we have \(b^2 = a^2(e^2 - 1)\). Taking the square root of both sides, we get \(b = \sqrt{a^2(e^2 - 1)}\). So, the length of the latus rectum is: \(l = 2b = 2\sqrt{a^2(e^2 - 1)}\).

Key Concepts

Latus Rectum

The latus rectum is a vital concept when it comes to understanding hyperbolas and other conic sections. Particularly, the latus rectum is the chord that passes through the focus of a conic section and is perpendicular to its major axis. It plays an important role in the geometry of conics.

For a hyperbola, the latus rectum is related specifically to the foci and the semi-major axis, defining how the curve extends around its center in a specific measure. Since a hyperbola has two foci, there are two segments of the latus rectum, each associated with one of its foci. To determine the length of the latus rectum of a hyperbola, it's crucial to evaluate it in terms of the hyperbola's parameters: the semi-major axis \(a\), semi-minor axis \(b\), and eccentricity \(e\).

In the case of a hyperbola centered at the origin, we use the equation:

• \(b^2 = a^2(e^2 - 1)\)
• Length of latus rectum \(= 2b\)

By substituting \(b\) into the equation \(l = 2\sqrt{a^2(e^2 - 1)}\), we finalize the length of the latus rectum, bringing in clear geometric information about the hyperbola's width at the focus.

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane. These include circles, ellipses, parabolas, and hyperbolas. They hold a critical place in both geometry and algebra, where each type of conic has distinct properties and equations.

Here's a quick overview of conic sections:

• Circles: Defined by the equation \(x^2 + y^2 = r^2\), where \(r\) is the radius.
• Ellipses: Have the general equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the semi-major and semi-minor axes, respectively.
• Parabolas: Characterized by the equation \(y^2 = 4ax\), where \(a\) determines the distance of the vertex from the focus.
• Hyperbolas: Defined by the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), with \(a\) and \(b\) determining the axes of the hyperbola.

Each conic section is determined by its eccentricity, which describes how "stretched" out the conic is. Despite their differences, they are all mathematically interrelated and have fascinating properties that provide solutions to various geometric problems.

Eccentricity

Eccentricity is a crucial parameter that distinguishes conic sections from each other. It effectively measures how much a conic section deviates from being circular. Each type of conic section is defined by a different range of eccentricity values, which alters its geometry significantly.

Here's a breakdown of eccentricity in each conic:

• Circle: Eccentricity \(e = 0\) since it is perfectly symmetrical in all directions.
• Ellipse: \(0 < e < 1\). The closer \(e\) is to zero, the more circular the ellipse is.
• Parabola: Eccentricity \(e = 1\) signifying its opening shape.
• Hyperbola: \(e > 1\), which means the two branches can stretch infinitely, apart from each other.

In hyperbolas specifically, eccentricity not only dictates the openness of the branches but also connects with the semi-major and semi-minor axes through the relation \(e = \frac{\sqrt{a^2 + b^2}}{a}\). This formula assists in calculating the semi-minor axis or eccentricity as required, providing significant insights into the structure of the hyperbola.

Semi-major Axis

In any conic section, the semi-major axis is one of the most significant geometric features. Specifically for hyperbolas, it represents half of the longest diameter that passes through the center. This term is most commonly used in ellipses, but it functions similarly in hyperbolas.

The semi-major axis \(a\) relates to various other components of the hyperbola:

• It is used in the standard equation of a hyperbola, as in \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) for horizontally oriented hyperbolas.
• It helps in determining the distance between the two vertices, with 2\(a\) being the full span of the major axis.
• The semi-major axis also connects to the eccentricity, allowing derivations of other key formulas within a hyperbola, linking to the focal distance by \(f = ae\).

By understanding the role of the semi-major axis, one can grasp many geometric and algebraic properties of hyperbolas, making it a cornerstone in the study of conic sections.