Problem 102 - College Algebra with Modeling and Visualization

Question

Given the standard equation of an ellipse, explain how to determine the length of the major axis. How can you determine whether the major axis is vertical or horizontal?

Step-by-Step Solution

Answer

Length of the major axis is twice the larger denominator's square root, and orientation is based on which variable it is under.

Step 1: Understanding the Standard Equation of the Ellipse

The standard equation of an ellipse looks like this: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \) or \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \), where \( (h, k) \) is the center of the ellipse. The values \(a\) and \(b\) are the semi-major and semi-minor axes, respectively. The exact role of \(a\) and \(b\) depends on which is larger.

Step 2: Determining the Length of the Major Axis

Compare the denominators \(a^2\) and \(b^2\). The major axis is the longest axis of the ellipse. If \(a > b\), then \(a\) is the length of the semi-major axis and the major axis has a length of \(2a\). If \(b > a\), then \(b\) is the semi-major axis and its length is \(2b\). Thus, the length of the major axis is always twice the larger of \(a\) or \(b\).

Step 3: Determining the Orientation of the Major Axis

To figure out if the major axis is vertical or horizontal, look at which variable (\(x\) or \(y\)) is associated with the larger denominator. If \(a > b\), the \(x\)-term has the larger denominator and the major axis is horizontal. If \(b > a\), the \(y\)-term has the larger denominator, indicating that the major axis is vertical.

Key Concepts

Standard Equation

An ellipse can be described by its standard equation, which takes the form of either \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \) or \( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \). Here, \((h, k)\) represents the center of the ellipse.
The constants \(a\) and \(b\) play crucial roles in defining the shape and size of the ellipse. Specifically, they refer to the lengths of the semi-major and semi-minor axes.
If you're looking at an ellipse and wondering which part of the equation relates to the bigger stretch or squish, compare \(a\) and \(b\). Whichever is larger indicates the direction where the ellipse extends the most.

Major Axis

The major axis of an ellipse is its longest diameter. To find the length of the major axis, you need to first identify whether \(a\) or \(b\) is greater in the standard equation.

• If \(a > b\), then \(a\) is the semi-major axis, and the major axis length is \(2a\).
• If \(b > a\), then \(b\) becomes the semi-major axis, and the major axis length is \(2b\).

It's important to note that the major axis is simply twice the length of the semi-major axis, making it straightforward once you recognize the larger value. This helps in understanding how wide or tall the ellipse stretches.

Orientation of Axes

Knowing whether the major axis is horizontal or vertical is key to understanding the orientation of an ellipse.
Look at the terms associated with the larger value in the standard equation. The term \((x - h)^2\) or \((y - k)^2\) will give you important clues:

• If \(a > b\), meaning the \(x\)-term's denominator is larger, the major axis is horizontal, stretching along the x-axis.
• If \(b > a\), meaning the \(y\)-term is dominant, the major axis is vertical, extending along the y-axis.

This orientation tells you the direction in which the ellipse is "longer," providing a visual sense of its dimensions in the coordinate plane.