Question

the graph shows an ellipse. write its equation in standard form.

Explanation:

Step1: Determine the center of the ellipse

The center \((h,k)\) of the ellipse is \((4,7)\) (from the graph).

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

Count the horizontal distance from the center to the ellipse. \(a = 3\) (since from \(x = 4\) to \(x=7\) or \(x = 1\)).

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

Count the vertical distance from the center to the ellipse. \(b = 2\) (since from \(y = 7\) to \(y = 5\) or \(y=9\)).

Step4: Write the standard form of the ellipse equation

The standard form of an ellipse with a horizontal major axis is \(\frac{(x - h)^2}{a^2}+\frac{(y - k)^2}{b^2}=1\). Substituting \(h = 4,k = 7,a = 3,b = 2\) gives \(\frac{(x - 4)^2}{9}+\frac{(y - 7)^2}{4}=1\).

Answer:

\(\frac{(x - 4)^2}{9}+\frac{(y - 7)^2}{4}=1\)