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Chapter 10: Problem 70

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{y^{2}}{9}-\frac{x^{2}}{1}=1\)

Short Answer

Expert verified
The similarity between the graphs of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{y^{2}}{9}-\frac{x^{2}}{1}=1\) is that both are hyperbolas with the same eccentricity. The difference is that the first hyperbola opens horizontally and the second hyperbola opens vertically.

Step by step solution

01

Identify the Types of Hyperbolas

Look at the given equations. Whenever the subtraction happens between \(x^{2}\) and \(y^{2}\), these are hyperbolas. Hence both equations represent hyperbolas.
02

Determine Orientation of Hyperbolas

The orientation of the hyperbola is determined by the signs. For the equation \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\), \(x^{2}\) is positive, so the hyperbola opens horizontally. For the equation \(\frac{y^{2}}{9}-\frac{x^{2}}{1}=1\), \(y^{2}\) is positive, so the hyperbola opens vertically.
03

Identify the Similarity and the Difference

The similarity is that both equations represent hyperbolas, and the hyperbolas have the same numbers in their denominators (9 for \(x^2\) or \(y^2\) and 1 for \(y^2\) or \(x^2\)) which means they have the same eccentricity. The difference is in the orientation of the hyperbolas - one opens horizontally while the other opens vertically.

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