Problem 5 - Algebra and Trigonometry Real Mathematics, Real People

Question

Consider the ellipse given by \(\frac{x^{2}}{2^{2}}+\frac{y^{2}}{8^{2}}=1.\) Is the major axis horizontal or vertical?

Step-by-Step Solution

Answer

The major axis of the ellipse is vertical.

Step 1: Identify the values of a and b

From the given ellipse equation \(\frac{x^{2}}{2^{2}}+\frac{y^{2}}{8^{2}}=1\), we can identify that \(a=2\) and \(b=8\).

Step 2: Compare the values of a and b

Since \(b=8\) is greater than \(a=2\), the major axis lies along the y-axis, which means this is a vertical ellipse.

Key Concepts

Major Axis

The major axis of an ellipse is a key feature that helps us understand the size and direction of the ellipse. It's the longest diameter of the ellipse and passes through the center, stretching to both sides to the edge. Imagine the major axis as a string spanning across the ellipse from one side to the opposite side, passing right through the middle of the ellipse. The major axis determines not just the size but also the orientation of the ellipse.
In mathematical terms:

• If the equation of the ellipse is \ \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \ \), then the major axis is along the variable with the larger denominator in the square, between \( x^2 \) and \( y^2 \).
• The major axis corresponds to the larger value between \( a \) and \( b \). If \( b > a \), the ellipse is vertically oriented, and vice versa.

Understanding the major axis can make it easier to visualize the shape of the ellipse and its alignment.

Horizontal ellipse

A horizontal ellipse has its major axis aligned parallel to the x-axis. This means that it stretches wider than it is tall. To quickly identify a horizontal ellipse in an equation:

• Check the terms in the equation formatted like \ \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \ \).
• If \( a > b \), the ellipse has a major axis along the x-axis, making it horizontal.

For a practical understanding, picture a sideways oval. The length that travels left to right is greater than the height from top to bottom. This would make the ellipse broad and flat.
Horizontal ellipses are typically encountered when looking at features that naturally spread out across a flat surface, like a running track viewed from above.

Vertical ellipse

A vertical ellipse, on the other hand, has its major axis aligned parallel to the y-axis. This means it is taller than it is wide. You can spot a vertical ellipse in the ellipse equation by:

• Noting the structure \ \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \ \).
• If \( b > a \), then the y-axis is the major axis, making it vertical.

Visualize it as an upright oval, more stretched from top to bottom. It resembles the shape of a standing egg.
This kind of ellipse is often encountered in diagrams featuring tall structures or objects. For example, consider how light might spread upwards in a lighthouse beam or look at a standing mirror.