Question

identify a and b for the ellipse with equation $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$.
a =
b =
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Explanation:

Step1: Analyze the ellipse graph

The standard form of an ellipse centered at the origin is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\), where \(a\) is the semi - major axis (the distance from the center to the vertex along the x - axis if \(a>b\)) and \(b\) is the semi - minor axis (the distance from the center to the vertex along the y - axis if \(a > b\)) or vice - versa. From the graph, the ellipse intersects the x - axis at \(x=\pm6\) and the y - axis at \(y = \pm2\).

Step2: Determine the values of \(a\) and \(b\)

For the ellipse equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), the x - intercepts are \((\pm a,0)\) and the y - intercepts are \((0,\pm b)\). Since the x - intercepts are at \(x=\pm6\), we have \(a = 6\) (because the distance from the center \((0,0)\) to the x - intercept is \(a\)). The y - intercepts are at \(y=\pm2\), so \(b = 2\) (the distance from the center \((0,0)\) to the y - intercept is \(b\)).

Answer:

\(a=\boxed{6}\), \(b=\boxed{2}\)