Problem 65 - College Algebra

Question

Use a graphing utility to graph \(\frac{x^{2}}{4}-\frac{y^{2}}{9}=0 .\) Is the graph a hyperbola? In general, what is the graph of \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0 ?\)

Step-by-Step Solution

Answer

The graph of \(\frac{x^{2}}{4}-\frac{y^{2}}{9}=0\) is a pair of intersecting lines, not a hyperbola. By generalization, the graph of \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0\) is a hyperbola when \(a ≠ b\) and a pair of intersecting lines when \(a = b\).

Step 1: Graph the given equation

Use a graphing utility which can handle equations of multiple variables to plot the equation \(\frac{x^{2}}{4}-\frac{y^{2}}{9}=0\). The graphing tool would plot points \((x, y)\) which satisfy this equation.

Step 2: Analyze the plot

Observe the graph created by the graphing utility. For the given equation, the graph will be a pair of intersecting lines, not a hyperbola. This is because the equation can be simplified as \(x^2 - y^2 = 0\), which further simplifies to \(x = ±y\), representing two intersecting lines instead of a hyperbola.

Step 3: Generalize to the form of the equation

In general terms, an equation \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0\) is a hyperbola when \(a ≠ b\) and a pair of intersecting lines when \(a = b\).