Question

find an equation for the ellipse section shown below.

Explanation:

Step1: Determine the center of the ellipse

The center of the ellipse \((h,k)\) is at the mid - point. Observing the graph, the center is at \((0,0)\) since it is symmetric about the \(x\) and \(y\) axes.

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

The distance from the center to the vertices along the \(y\) - axis: The top vertex is at \((0,6)\) and the bottom vertex is at \((0, - 4)\). The length of the major axis \(2a\) (assuming the major axis is vertical). The distance from the center \((0,0)\) to the top vertex \((0,6)\) gives \(a = 6\). The distance from the center \((0,0)\) to the right - hand side of the ellipse (along the \(x\) - axis): The right - hand side point is at \((3,0)\), so \(b = 3\)

Step3: Use the standard form of the ellipse equation

The standard form of an ellipse with center \((h,k)=(0,0)\) and major axis vertical is \(\frac{(x - h)^{2}}{b^{2}}+\frac{(y - k)^{2}}{a^{2}} = 1\)
Substitute \(h = 0,k = 0,a = 6,b = 3\) into the equation:

$$
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Answer:

\(\frac{x^{2}}{9}+\frac{y^{2}}{36}=1\)