Problem 58 - Algebra and Trigonometry

Question

How Wide Is an Ellipse at Its Foci? 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 the latus rectum is \( \frac{2b^{2}}{a} \).

Step 1: Understand the Problem

We are given an ellipse with the equation \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) where \( a > b \). We need to find the length of the latus rectum, which is a line segment perpendicular to the major axis at a focus of the ellipse.

Step 2: Coordinate of the Focus

For the ellipse \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \), the focal distance \( c \) is given by \( c = \sqrt{a^{2} - b^{2}} \). Therefore, the coordinates of the foci are \(( \pm c, 0) = (\pm \sqrt{a^{2} - b^{2}}, 0)\).

Step 3: Equation of the Latus Rectum

The latus rectum is a vertical line that is perpendicular to the major axis, passing through the focus. We consider the line \( x = c \) (one of the foci). The intersection of this line with the ellipse gives us the endpoints of the latus rectum.

Step 4: Substitute x=c into the Ellipse Equation

Substitute \( x = \sqrt{a^{2} - b^{2}} \) into the ellipse equation \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \). Simplifying gives:\[ \frac{(a^{2} - b^{2})}{a^{2}} + \frac{y^{2}}{b^{2}} = 1.\]

Step 5: Solve for y^2

Rearrange the equation to solve for \( y^{2} \):\[ \frac{y^{2}}{b^{2}} = 1 - \frac{(a^{2} - b^{2})}{a^{2}} = \frac{b^{2}}{a^{2}}.\]This implies \( y^{2} = \frac{b^{4}}{a^{2}} \) so \( y = \pm \frac{b^{2}}{a} \).

Step 6: Length of the Latus Rectum

The length of the latus rectum is the distance between the points \( (c, -\frac{b^{2}}{a}) \) and \( (c, \frac{b^{2}}{a}) \) along the y-axis. This distance is:\[\text{Length} = \left( \frac{b^{2}}{a} - \left( -\frac{b^{2}}{a} \right) \right) = 2 \times \frac{b^{2}}{a} = \frac{2b^{2}}{a}.\]

Step 7: Conclusion

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

Key Concepts

Ellipse Equation

An ellipse is a fascinating shape in geometry, defined by a specific algebraic equation. For an ellipse centered at the origin with its major axis along the x-axis, the standard equation is:

• \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \)

Here, \(a\) and \(b\) are constants that represent the semi-major and semi-minor axes of the ellipse, respectively. The larger value among \(a\) and \(b\) determines the direction of the major axis. In our context, \(a > b\), indicating that the major axis runs parallel to the x-axis.
This equation is significant because it provides all the information needed to understand the ellipse's shape and orientation.
To visualize an ellipse, imagine stretching a circle along one axis more than the other. This stretched circle is our ellipse. Its proportions and orientation are entirely governed by our ellipse equation.

Latus Rectum

The latus rectum is a unique feature of an ellipse, essential to understanding its geometry. It refers to a line segment that is perpendicular to the major axis and passes through one of the foci of the ellipse.

Think of the latus rectum as a measure of the "width" of the ellipse at its focus. For an ellipse, this line segment helps illustrate the curvature near the foci, providing insights into how "stretched" the ellipse is around its foci. More specifically, the latus rectum ends at points on the ellipse itself.
In the problem presented, the length of the latus rectum is always calculated using a similar process, where the endpoints are determined by intersecting a vertical line with the ellipse equation, precisely at the focus. This calculation provides an exact understanding of its length, which fortifies the complete understanding of ellipse geometry.

Focal Distance

The focal distance is a crucial element in defining an ellipse. It helps us to locate the focuses (foci) of the ellipse along its major axis. The foci are two points essential in the construction and understanding of the ellipse's properties.

The formula to find the focal distance \(c\) for an ellipse defined by the equation \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) is:

• \( c = \sqrt{a^{2} - b^{2}} \)

This formula lets us calculate the distance from the center of the ellipse to either of the foci placed symmetrically along the major axis. This center-to-focus distance points towards our ellipse's elongated nature on the major axis.
Once found, these focal points \(( \pm c, 0)\) underline the symmetry of the ellipse, shedding light on both its form and mathematical properties crucial for further analyses.

Length Calculation

Calculating the length of the latus rectum involves a few straightforward steps, beginning with understanding how it interacts with the ellipse's geometry. To solve for this length, follow these steps:

1. **Substitute**: First, substitute the coordinate of one of the foci \(x = c\) into the ellipse equation.2. **Solve for \(y^{2}\)**: This involves manipulating the equation: \[ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \] becomes \[ \frac{(a^{2} - b^{2})}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \] which simplifies to solve \(y^{2} = \frac{b^{4}}{a^{2}}\).3. **Calculate \(y\)**: The solutions for \(y\) are \(y = \pm \frac{b^{2}}{a}\). These represent the vertical positions where the line intersects the ellipse.4. **Find the full length**: Finally, calculate the distance between these two intersection points: \[ \text{Length} = \frac{b^{2}}{a} - \left(- \frac{b^{2}}{a}\right) = 2 \times \frac{b^{2}}{a} = \frac{2b^{2}}{a}\]This length is not just a number; it provides substantial insight into how broad the ellipse appears at its foci, revealing how the shape dynamically stretches at these critical points.