Problem 70 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.