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Chapter 7: Problem 65

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 \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0 ?\)

Short Answer

Expert verified
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 \(a 鈮� b\) and a pair of intersecting lines when \(a = b\).

Step by step solution

01

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.
02

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 \(x = 卤y\), representing two intersecting lines instead of a hyperbola.
03

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 \(a 鈮� b\) and a pair of intersecting lines when \(a = b\).

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Most popular questions from this chapter

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of its foci. $$4 x^{2}+y^{2}+16 x-6 y-39=0$$

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