Question

identify a and b for the hyperbola with equation $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1$.

Explanation:

Step1: Recall the standard form of hyperbola

The standard form of a hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} = 1\) has vertices at \((\pm a,0)\) and asymptotes \(y=\pm\frac{b}{a}x\).

Step2: Determine the value of \(a\)

From the graph, the vertices of the hyperbola are at \((\pm6,0)\). In the standard form \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\), when \(y = 0\), \(x=\pm a\). So \(a = 6\) (already given correctly).

Step3: Determine the value of \(b\)

The asymptotes of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) are \(y=\pm\frac{b}{a}x\). From the graph, the asymptotes pass through the points that suggest the slope of the asymptotes. Since \(a = 6\) and the asymptotes have a slope of \(1\) (because for example, if we consider the line \(y = x\) as an asymptote, and \(y=\frac{b}{a}x\), then \(\frac{b}{a}=1\)). Substituting \(a = 6\) into \(\frac{b}{a}=1\), we get \(b=a\).

Answer:

\(b = 6\)