Problem 77 - Algebra and Trigonometry

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 the equation \( \frac{x^{2}}{4}-\frac{y^{2}}{9}=0 \) is not a hyperbola, instead, it is a pair of intersecting lines. Similarly, the graph of the equation \( \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0 \) for any values of a and b, is a pair of intersecting lines too, and not a hyperbola.

Step 1: Graph the Function

Input the equation \( \frac{x^{2}}{4}-\frac{y^{2}}{9}=0 \) into a graphing utility. Observe the shape of the graph it generates.

Step 2: Identify the Conic Section

Analyzing the graph, it represents a line and not a hyperbola. This is due to the fact that the equation of a hyperbola has subtraction between the two terms but also the right side of the equation is usually a positive number other than zero.

Step 3: Generalize the Graph

The general form of the equation mentions is \( \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0 \). If graphed, this equation won't give a typical hyperbola. Instead, it will represent a pair of lines. This is because the equation could be rewritten as \( (\frac{x}{a})^2 = (\frac{y}{b})^2 \), which simplifies to \( \frac{x}{a}=\frac{y}{b} \) and \( \frac{x}{a}=-\frac{y}{b} \), forming a pair of intersecting lines along the axes.