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 centered at the origin of the coordinate system).

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

The distance from the center to the top (or bottom) of the ellipse (along the \(y\) - axis). Looking at the graph, if we count the units from the center \((0,0)\) to the top of the ellipse, \(a = 10\).

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

The distance from the center to the right (or left) of the ellipse (along the \(x\) - axis). Looking at the graph, if we count the units from the center \((0,0)\) to the right of the ellipse, \(b=3\).

Step4: Use the standard form of the ellipse equation

The standard form of an ellipse centered at \((h,k)\) is \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\) (since it is a vertical ellipse, the \(y\) - term has the larger denominator). Substituting \(h = 0,k = 0,a = 10,b = 3\) into the formula, we get \(\frac{x^{2}}{9}+\frac{y^{2}}{100}=1\).

Answer:

\(\frac{x^{2}}{9}+\frac{y^{2}}{100}=1\)