Question

give the equation for the ellipse graphed above.

Explanation:

Step1: Determine the center of the ellipse

The center of the ellipse \((h,k)\) is at \((0,0)\) (since it is symmetric about the origin).

Step2: Find the value of \(a\) (semi - major axis)

The distance from the center to the vertex along the \(y\) - axis. The vertices are at \((0,9)\) and \((0, - 9)\), so \(a = 9\).

Step3: Find the value of \(b\) (semi - minor axis)

The distance from the center to the co - vertex along the \(x\) - axis. The co - vertices are at \((3,0)\) and \((-3,0)\), so \(b = 3\).

Step4: Write the standard form of the ellipse equation

The standard form of an ellipse centered at \((h,k)\) is \(\frac{(x - h)^{2}}{b^{2}}+\frac{(y - k)^{2}}{a^{2}}=1\). Substituting \(h = 0,k = 0,a = 9,b = 3\) into the formula, we get \(\frac{x^{2}}{3^{2}}+\frac{y^{2}}{9^{2}}=1\).

Answer:

\(\frac{x^{2}}{9}+\frac{y^{2}}{81}=1\)