Question

In Problems 4–6, find an equation of the conic described; graph the equation.

Ellipse: center$\left(0,0\right)$, vertex$\left(0,-4\right)$, focus$\left(0,3\right)$

Short answer

The equation of the ellipse is $\frac{x^2}{7}+\frac{y^2}{16}=1$

The graph of the equation is

Step-by-step solution

Step 1. Given Information

Given to determine an equation of the conic described by

Ellipse: center$\left(0,0\right)$, vertex$\left(0,-4\right)$, focus$\left(0,3\right)$

The equation is to be graphed.

Step 2. Determining the equation

Here, the center of the ellipse is Origin and the given focus and vertex lie on the y-axis. Hence the major axis of the ellipse is the y-axis.

The distance from the center of the ellipse i.e. origin to the focus given is $c=3$

The distance from the center of the ellipse i.e. origin to the vertex given is $a=4$

For an ellipse with major axis along y-axis, role="math" $b^2=a^2-c^2$

Plugging the values:

$$\begin{gathered} b^2=a^2-c^2 \\ {b}^2=4^2-3^2 \\ {b}^2=16-9 \\ {b}^2=7 \end{gathered}$$

The equation of an ellipse with major axis along y-axis is $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$

Plugging the values:

$$\begin{gathered} \frac{x^2}{b^2}+\frac{y^2}{a^2}=1 \\ \frac{x^2}{7}+\frac{y^2}{16}=1 \end{gathered}$$

Hence the equation of the ellipse is$\frac{x^2}{7}+\frac{y^2}{16}=1$

Step 3. Graph of the equation

Graphing the equation using graphing utility:

Step 4. Conclusion

The equation of the ellipse is $\frac{x^2}{7}+\frac{y^2}{16}=1$

The graph of the equation is