Problem 58 - Precalculus Mathematics for Calculus

Question

A latus rectum for an ellipse is a line segment perpendicular to the major axis at a focus, with endpoints on the ellipse, as shown in the figure at the top of the next column. Show that the length of a latus rectum is \(2 b^{2} / a\) for the ellipse.$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad \text { with } a>b$$

Step-by-Step Solution

Answer

The length of a latus rectum for the ellipse is \(\frac{2b^2}{a}\).

Step 1: Understand the Focus and Latus Rectum Position

To determine the length of the latus rectum, you first need to understand its position. The length you're finding is a line segment that passes through one of the foci of the ellipse and is perpendicular to the major axis. This segment's endpoints lie on the ellipse.

Step 2: Identify the Ellipse Components

Given the ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the semi-major axis length is \(a\) and semi-minor axis length is \(b\). The distance from the center to each focus is \(c\), where \(c = \sqrt{a^2 - b^2}\).

Step 3: Set Up the Latus Rectum Equation

The latus rectum is a vertical line when the major axis is horizontal as in this ellipse. Therefore, for a given focus \((c, 0)\), it occurs through the focus and has equal lengths above and below the x-axis due to symmetry.

Step 4: Use the Ellipse Equation to Find Latus Rectum Intersection

The latus rectum is at \(x = c\). Substitute \(x = c = \sqrt{a^2 - b^2}\) into the ellipse equation: \[\frac{c^2}{a^2} + \frac{y^2}{b^2} = 1\].

Step 5: Solve for the y-values

With \(c = \sqrt{a^2 - b^2}\), substitute it in: \[\frac{a^2 - b^2}{a^2} + \frac{y^2}{b^2} = 1\]. Simplify to find \(y^2\):

Step 6: Simplify to Get Latus Rectum Length

Re-arrange and simplify to find \(y^2 = \frac{b^4}{a^2}\), implying that the length of the latus rectum is twice this value, \[y = \pm \frac{b^2}{a}\].

Step 7: Conclude with the Length of the Latus Rectum

The total length of the latus rectum, considering both directions (above and below the x-axis), is \(2 \cdot \frac{b^2}{a}\). Therefore, the length is \(\frac{2b^2}{a}\).

Key Concepts

Ellipse Equation

An important start to understanding ellipses is their equation, which in its standard form is given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a\) and \(b\) are real numbers where \(a > b\). This equation defines the set of all points \((x, y)\) that form the shape of an ellipse on a coordinate plane.
The terms in the equation can tell you a lot about the ellipse's shape:

• \(a^2\) is associated with the x-term and represents the semi-major axis squared if \(a > b\).
• \(b^2\) is associated with the y-term and signifies the semi-minor axis squared.

Therefore, the ellipse centered at the origin \((0, 0)\) has a horizontal orientation if \(a > b\), putting the longer dimension along the x-axis. This is key when considering the ellipse's structure and the lengths and intersections of lines like the latus rectum.

Major and Minor Axes

In an ellipse, the major and minor axes are critical components that determine its shape. The **major axis** is the longest diameter and runs along the broadest part of the ellipse, while the **minor axis** is the shortest diameter. In mathematical equations of ellipses, if \(a > b\) then:

• The length of the major axis is \(2a\), representing the full span across the ellipse's maximum width.
• The length of the minor axis is \(2b\), representing the full span across the ellipse's minimum width.

The intersections of these axes are at the center of the ellipse. The major axis provides a line of symmetry, dividing the ellipse into two equal halves. Knowing the axes helps to pinpoint features like the foci, which are essential for solving problems related to ellipses, such as finding the latus rectum.

Foci of an Ellipse

The foci (plural of focus) are two specific points inside an ellipse that have special significance. They are positioned symmetrically along the major axis, helping define the ellipse's shape. For an ellipse with the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the distance to the foci from the center is given by \(c\), calculated with \(c = \sqrt{a^2 - b^2}\).
These points have a unique property—the sum of the distances from any point on the ellipse to the two foci is constant. Here's a quick look at their significance:

• They provide a way to find other geometric properties, such as the latus rectum.
• The position \((\pm c, 0)\) is crucial in solving the equations related to the ellipse structure.

The understanding of foci allows you to analyze how the ellipse stretches or compresses. They essentially act as anchors for the ellipse's symmetric properties.

Conic Sections

Ellipses are one of the four types of conic sections, which are the figures formed by slicing a double-napped cone at different angles. The other types include parabolas, hyperbolas, and circles. Ellipses form when the plane cuts through the cone at an angle, but doesn't intersect the base. This concept offers further insight into the nature of ellipses:

• Conic sections are fundamental in mathematics and upper-level geometry.
• Ellipses can be visualized as stretching circles, incorporating both major and minor axes.

Conic sections connect to various applications, from planetary orbits to engineering. Understanding ellipses as conic sections not only allows one to solve mathematical problems but also to relate these geometric figures to real-world contexts.