Question

Find an equation for each ellipse. Graph the equation by hand.

Center at$(0,0)$, vertex at$(0,4),b=1$

Short answer

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

Step-by-step solution

Step 1. Given Information

We are given vertex at$(0,4),b=1$

Step 2. Calculation

The vertices lie on the y-axis. So, the major axis of the ellipse is y-axis and the vertices formed by minor axis are $(1,0)and(-1,0)$

Find the value of foci using the formula.

$$\begin{gathered} b^2=a^2-c^2 \\ {1}^2=4^2-c^2 \\ {c}^2=16-1 \\ {c}^2=15 \end{gathered}$$

Thus, the equation of ellipse is as follows,

$$\begin{gathered} \frac{x^2}{b^2}+\frac{y^2}{a^2}=1 \\ \frac{x^2}{1^2}+\frac{y^2}{4^2}=1 \\ {x}^2+\frac{y^2}{16}=1 \end{gathered}$$

On plotting the points on the graph, we get,