Chapter 9.2, Problem 17E

To determine

To find:The standard equation of the ellipse with center at origin when foci are $\left(0,\pm 5\right)$ and the length of major axis is $14$ .

Answer to Problem 17E

The equation of the ellipse is $\frac{x^2}{24}+\frac{y^2}{49}=1$ .

Explanation of Solution

Given information:

The foci of the ellipse is $\left(0,\pm 5\right)$ andthe length of major axis is $14$ .

Calculation:

The major axis of the ellipse is vertical as the foci is lying on the y-axis. So, the general equation of ellipse is $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ .

The length of major axis is $14$ .

  $\begin{array}{l}2a=14\\ a=7\end{array}$

Substitute $7$ for $a$ and $5$ for $c$ in the formula.

  $\begin{array}{c}{c}^2=a^2-b^2\\ 5^2=7^2-b^2\\ b^2=49-25\\ b^2=24\end{array}$ .

Substitute the values in the general equation of the ellipse.

  $\begin{array}{c}\frac{x^2}{b^2}+\frac{y^2}{a^2}=1\\ \frac{x^2}{24}+\frac{y^2}{49}=1\end{array}$

Therefore, the equation of the ellipse is $\frac{x^2}{24}+\frac{y^2}{49}=1$ .