Problem 71 - Algebra 2

Question

What is the length of the minor axis of the graph of \(\frac{x^{2}}{100}+\frac{y^{2}}{64}=1 ?\) \(\begin{array}{llll}{\text { A. } 12} & {\text { B. } 2 \sqrt{41}} & {\text { C. } 16} & {\text { D. } 20}\end{array}\)

Step-by-Step Solution

Answer

The minor axis is 16 units long. Therefore, the right choice is C. 16.

Step 1: Identifying the semi-minor axis

Look at the equation of the ellipse which is \(\frac{x^{2}}{100} + \frac{y^{2}}{64} = 1\). The denominator under the y-term (64) is a square of the length of semi-minor axis. So, the semi-minor axis is the square root of 64, which equals 8.

Step 2: Finding the minor axis

The minor axis is twice the length of the semi-minor axis. So, the length of the minor axis is 2*8 = 16.

Key Concepts

Minor Axis

The minor axis of an ellipse is an essential feature that helps define its shape and orientation. It is the shortest diameter of the ellipse and is perpendicular to the major axis, which is the longest diameter. To understand its significance, remember:

• In the standard form of an ellipse equation, the minor axis aligns with the term that has the smaller denominator.
• The length of the minor axis is twice the length of the semi-minor axis, providing a complete measure across the ellipse.
• For the equation given by \[ \frac{x^{2}}{100} + \frac{y^{2}}{64} = 1 \]the value under the \(y^2\) term corresponds to the semi-minor axis, indicating the minor axis extends in the direction of the y-axis.

A simple computation reveals that the minor axis here has a length of 16, making it key to understanding the ellipse's full dimensions.

Semi-Minor Axis

The semi-minor axis of an ellipse is half of the minor axis and is essential in determining the ellipse's contour. When looking at the ellipse's equation,\[\frac{x^{2}}{100} + \frac{y^{2}}{64} = 1,\]you focus on the terms under the square roots. Here’s how to identify the semi-minor axis:

• The semi-minor axis is associated with the denominator of the \(y^{2}\) term since 64 is less than 100.
• To find its value, compute the square root of the smaller denominator, \(\sqrt{64} = 8\), which yields the length of the semi-minor axis.
• This axis is crucial for plotting and understanding the vertical span of the ellipse.

Understanding the semi-minor axis helps set up and visualize how the ellipse stretches along its shorter dimension.

Equation of Ellipse

The equation of an ellipse, in its standard form, is pivotal in understanding its dimensions and orientation. The general structure is:\[\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\]where \(a\) and \(b\) correspond to the semi-major and semi-minor axes, respectively. Key considerations include:

• If \(a > b\), the ellipse is stretched along the x-axis with a horizontal orientation.
• If \(b > a\), it is wider on the y-axis, giving a vertical orientation.
• The denominators \(a^2\) and \(b^2\) are directly tied to the square of the semi-major and semi-minor axes, making these numbers crucial in calculating their lengths.

In the given equation, \(\frac{x^{2}}{100} + \frac{y^{2}}{64} = 1\), \(b^2 = 64\) and \(a^2 = 100\). This signifies that this particular ellipse is horizontally arranged, emphasizing the practical usability of the equation format in predicting the ellipse's geometry.