Chapter 11, Problem 9RE

To determine

To find: The center, vertices, foci and the lengths of the major and minor axes of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and draw its graph.

Answer to Problem 9RE

The center, vertices, foci and the lengths of the major and minor axes of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ are $\left(0,0\right)$ , $\left(0,\pm 3\right)$ , $\left(0,\pm 4\right)$ , 10 and 6, respectively.

Explanation of Solution

Given information:

The given equation is $\frac{x^2}{9}+\frac{y^2}{25}=1$ .

Calculation:

Write the equation in standard form of ellipse.

  $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$

Compare the given equation with standard form of equation of ellipse.

  $a=5$ , $b=3$

Here, $b>a>0$ . The center is at origin.

Calculate the vertices of the ellipse.

  $\left(0,\pm a\right)=\left(0,\pm 3\right)$

Calculate the foci of the ellipse.

  $\begin{array}{c}{c}^2=5^2-3^2\\ c^2=16\\ c=4\end{array}$

The foci of the ellipse is,

  $\left(0,\pm c\right)=\left(0,\pm 4\right)$

Calculate the length of the minor axis.

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

Calculate the length of the major axis.

  $\begin{array}{c}2a=2\times 5\\ =10\end{array}$

Draw the graph of the ellipse using graphical utility.