Problem 78 - Algebra and Trigonometry

Question

Graph \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) and \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\) in the same viewing rectangle for values of \(a^{2}\) and \(b^{2}\) of your choice. Describe the relationship between the two graphs.

Step-by-Step Solution

Answer

The two graphs are both hyperbolas which intersect at the x- and y-axes. They open in different directions: one along the x-axis and the other along the y-axis. They have the same semi-axes lengths and share the same foci. This creates a symmetric figure with respect to the origin and both axes.

Step 1: Graph the first equation

Choose some values for \(a^{2}\) and \(b^{2}\). These values will determine the shape of the hyperbola. For example, set \(a^{2} = 4\) and \(b^{2} = 1\). Then plug these values into the first equation and graph the resulting function, which is \(\frac{x^{2}}{4}-y^{2}=1\). The shape of the hyperbola is primarily determined by these values.

Step 2: Graph the second equation

Use the same values for \(a^{2}\) and \(b^{2}\) that were chosen for the first equation. This time plug these values into the second equation and graph the resulting function, which is \(\frac{x^{2}}{4}-y^{2}=-1\). This will yield a different hyperbola than the one obtained in step 1.

Step 3: Analyze the relationship between the graphs

Observe the two graphs and analyze their relationship. Since the two hyperbolas are derived from the same values of \(a^{2}\) and \(b^{2}\), they will intersect at the x- and y-axes, forming a symmetrical figure with respect to both the origin and the coordinate axes. The hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) will open along the x-axis, whereas \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1\) will open along the y-axis. Both hyperbolas will have the same semi-axes lengths, and they will also share the same foci.