Chapter 9.2, Problem 24E

To determine

To find:The standard equation of the ellipse when the vertices are $\left(3,1\right)$ , $\left(3,9\right)$ and length of minor axis is $6$ .

Answer to Problem 24E

The equation of the ellipse is $\frac{{\left(x-3\right)}^2}{3^2}+\frac{{\left(y-5\right)}^2}{4^2}=1$ .

Explanation of Solution

Given information:

The vertices of the ellipse are $\left(3,1\right)$ , $\left(3,9\right)$ and length of minor axis is $6$ .

Calculation:

The vertices of the ellipse are $\left(3,1\right)$ and $\left(3,9\right)$ .

Therefore, the center of the ellipse is at midpoint of the vertices.

  $\frac{1}{2}\left[\left(3,1\right)+\left(3,9\right)\right]=\left(3,5\right)$

So, the center of the ellipse is the point $\left(3,5\right)$ .

The length of the major axis is the distance between the points $\left(3,1\right)$ and $\left(3,9\right)$ .

  $\begin{array}{c}\sqrt{{\left(3-3\right)}^2+{\left(9-1\right)}^2}=\sqrt{8^2}\\ =8\end{array}$

The length of the minor axis is $6$ .

The major axis is vertical. So, the general equation of the ellipse is $\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1$ .

Use the formula of length of major axis.

  $\begin{array}{c}2a=8\\ a=4\end{array}$

Use the formula of length of minor axis.

  $\begin{array}{c}2b=6\\ b=3\end{array}$

Substitute $4$ for $a$ , $3$ for $b$ , $3$ for $h$ and $5$ for $k$ in the general equation of the ellipse.

  $\begin{array}{c}\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1\\ \frac{{\left(x-3\right)}^2}{3^2}+\frac{{\left(y-5\right)}^2}{4^2}=1\end{array}$

Therefore, the equation of the ellipse is $\frac{{\left(x-3\right)}^2}{3^2}+\frac{{\left(y-5\right)}^2}{4^2}=1$ .