Problem 65 - Intermediate Algebra

Question

Compare the graphs of \(\frac{x^{2}}{81}+\frac{y^{2}}{64}=1\) and \(\frac{x^{2}}{64}+\frac{y^{2}}{81}=1 .\) Do they have any similarities?

Step-by-Step Solution

Answer

The ellipses have the same dimensions but are rotated 90 degrees relative to each other.

Step 1: Identify the Type of Graphs

Both equations are in the form of the standard equation for an ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). The first graph \(\frac{x^2}{81} + \frac{y^2}{64} = 1\) has \(a^2 = 81\) and \(b^2 = 64\), while the second graph \(\frac{x^2}{64} + \frac{y^2}{81} = 1\) has \(a^2 = 64\) and \(b^2 = 81\).

Step 2: Calculate Semi-Major and Semi-Minor Axes

For the first equation, the semi-major axis \(a = \sqrt{81} = 9\) and the semi-minor axis \(b = \sqrt{64} = 8\). For the second equation, the values are reversed, so the semi-major axis \(b = \sqrt{81} = 9\) and the semi-minor axis \(a = \sqrt{64} = 8\).

Step 3: Determine Orientation of Ellipses

The first ellipse, \(\frac{x^{2}}{81} + \frac{y^{2}}{64} = 1\), is elongated along the x-axis because its semi-major axis is aligned with the x-axis. The second ellipse, \(\frac{x^{2}}{64} + \frac{y^{2}}{81} = 1\), is elongated along the y-axis because its semi-major axis is aligned with the y-axis.

Step 4: Conclusion on Similarities

Both ellipses have the same semi-major and semi-minor axes, but they are oriented differently. The differences in orientation mean they are not identical but are related by a 90-degree rotation about the origin.

Key Concepts

Graphing Ellipses

An ellipse is a beautiful geometric shape that looks like an elongated circle. To draw an ellipse, you start with its standard equation: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]If you have this equation, you're on your way to understanding its shape and position.

• The terms \(x^2/a^2\) and \(y^2/b^2\) relate to the coordinates of the ellipse.
• The numbers \(a^2\) and \(b^2\) will help determine how 'stretched' your ellipse is.

To graph, plot a few key points:

• Where the ellipse crosses the axes
• Points halfway along the axes

Sketch smooth, curved lines between these points to complete the ellipse. Remember, all points should make the equation true when substituted back in. Trying it out can feel like connecting dots but with a gentle curve!

Semi-Major Axis

The semi-major axis is a crucial part of the ellipse. It is the longest radius from the center to the edge, showing how wide the ellipse is. In an ellipse with the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\),

• The semi-major axis is either \(a\) or \(b\), depending on which is larger.
• If \(a > b\), the ellipsis stretches more along the x-axis, \(a\), determining its length.
• If \(b > a\), it stretches along the y-axis, \(b\), determining its height.

To calculate the semi-major axis, take the square root of the larger denominator.
For example, if \(a^2 = 81\), then \(a = \sqrt{81} = 9\). This means on the graph, you measure from the center by 9 units in the direction of the corresponding axis.

Semi-Minor Axis

The semi-minor axis is equally important to understand the shape of an ellipse. This axis is the shorter radius, highlighting how 'narrow' the ellipse is. In an equation like \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\),

• The semi-minor axis is the smaller quantity between \(a\) (\(a = \sqrt{a^2}\)) and \(b\) (\(b = \sqrt{b^2}\)).
• If \(a < b\), \(a\) is your semi-minor axis, indicating the shorter curve on the x-axis.
• If \(b < a\), \(b\) is your semi-minor axis, indicating the shorter curve on the y-axis.

To find this shorter axis, take the square root of the smaller denominator from the equation.
This gives you the length from the center to the edge along that axis. Remember, the semi-minor axis helps complete the touch of elegance in the ellipse, defining its narrow width.

Orientation of Ellipses

An ellipse's orientation is all about which way it stretches more. Is it stretched sideways or up and down? Understanding this helps in drawing accurate representations. In the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\):

• If \(a > b\), the ellipse stretches more along the x-axis, lying horizontally.
• If \(b > a\), it stretches more along the y-axis, standing vertically.

For example, the equation \(\frac{x^2}{81} + \frac{y^2}{64} = 1\) stretches horizontally since \(81 > 64\) (making the x-direction longer).
Whereas, \(\frac{x^2}{64} + \frac{y^2}{81} = 1\) stretches vertically since \(81 > 64\) for the y-direction. Knowing the orientation helps in sketching because it gives you a clear idea of how to extend the curves.