Problem 70 - Algebra 2

Question

What is the length of the major axis on the graph of \(\frac{x^{2}}{100}+\frac{y^{2}}{64}=1 ?\) \(\begin{array}{llll}{\text { F. } 12} & {\text { G. } 2 \sqrt{41}} & {\text { H. } 16} & {\text { J. } 20}\end{array}\)

Step-by-Step Solution

Answer

The length of the major axis on the graph of the given equation is 20.

Step 1: Identify the values of 'a' and 'b'

The ellipse has the standard form \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\). Here, \(a^{2} = 100\) and \(b^{2} = 64\). Therefore, \(a = 10\) and \(b = 8\).

Step 2: Determine the length of the major axis

The length of the major axis is 2a if \(a>b\), or 2b if \(b>a\). Since \(a>b\) here, the length of the major axis is \(2 * a = 20\).

Key Concepts

Major Axis

In an ellipse, the major axis is the longest diameter that runs through the center and the two farthest points on the ellipse's boundary. It is crucial in determining the overall shape of the ellipse. The major axis will always be longer than the minor axis, which is the shortest line running through the center. To find the length of the major axis in a given ellipse graph, you determine which axis—horizontal or vertical—is longer. If the equation of the ellipse is of the form:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) with \(a > b\), the major axis is horizontal.
• Conversely, if \(b > a\), it is vertical.

The formula for the major axis length in standard orientation ellipse is simply twice the length of 'a' or 'b', whichever is larger. Therefore, in our exercise, the major axis is calculated by taking \(2a\), leading to a length of 20 units.

Standard Form of Ellipse

The standard form of an ellipse equation provides a simple yet powerful way to identify key characteristics quickly. It’s structured as follows: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where:

• \(a\) and \(b\) represent the semi-major and semi-minor axes respectively.
• The ellipse is centered at the origin unless different coordinates are specified.

This equation reveals which of the axes (horizontal or vertical) is longer, thus helping to ascertain the orientation of the ellipse. In the problem you encountered, \(a^2 = 100\) and \(b^2 = 64\) align with the formula, meaning that \(a = 10\) and \(b = 8\). Because \(a > b\), the ellipse's major axis is horizontal.

Axis Length Calculation

The axis length calculation is an essential step while engaging with ellipses. In our context, it involves establishing whether the given half-lengths (\(a\) and \(b\)) correspond to the horizontal or vertical axes, and then calculating the full lengths of each axis. Simplifying it:

• For the length of the major axis: Multiply the larger value (either \(a\) or \(b\)) by 2. This is denoted as \(2a\) or \(2b\).
• For the length of the minor axis: Multiply the smaller value (again, \(a\) or \(b\)) by 2.

In the exercise, \(a = 10\) is the larger value, thus the major axis length calculated is \(20\) units. Similarly, the minor axis, with \(b = 8\), would measure \(16\) units. Understanding these calculations helps in visualizing the ellipse and predicting its proportions accurately.