Question

find the equation of the ellipse with the following properties. express your answer in standard form. centered at (1,3) major axis of length 18 oriented vertically minor axis of length 2

Explanation:

Step1: Recall the standard form of an ellipse

The standard form of an ellipse centered at \((h,k)\) with a vertical major axis is \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\), where \(a\) is the semi - major axis length and \(b\) is the semi - minor axis length.

Step2: Determine the values of \(h,k,a,b\)

Given the center \((h,k)=(1,3)\).
Since the major axis length \(2a = 18\), then \(a=\frac{18}{2}=9\).
Since the minor axis length \(2b = 2\), then \(b=\frac{2}{2}=1\).

Step3: Substitute the values into the standard form

Substitute \(h = 1,k = 3,a = 9,b = 1\) into \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\).
We get \(\frac{(x - 1)^2}{1^2}+\frac{(y - 3)^2}{9^2}=1\), which simplifies to \(\frac{(x - 1)^2}{1}+\frac{(y - 3)^2}{81}=1\).

Answer:

\(\frac{(x - 1)^2}{1}+\frac{(y - 3)^2}{81}=1\)