Question

Write an equation for each ellipse.

Short answer

The equation of ellipse is$\frac{{(x+1)}^2}{4}+{(y-1)}^2=1$

Step-by-step solution

Step 1. Given Information 

The given graph is

Step 2. Calculation 

The ellipse in the given graph has center at the point $(-1,1)$So, role="math" localid="1647278093172" $h=-1,k=1$From the graph, we can say that the center, focus and vertex of ellipse all lie on the same line $y=1$. So, major axis is parallel to x-axis.

The vertices of the major axis of the ellipse are at $(h-a,k)=(-3,1)and(h+a,k)=(1,1)$and vertices of the minor axis are at

$(h,k-b)=(-1,0)and(h,k+b)=(-1,2)$

Consider $h+a=1andk+b=2$. Since $h=-1,k=1$, the equations will be as follows,

$$\begin{gathered} -1+a=1 \\ a=2 \\ and \\ 1+b=2 \\ b=1 \end{gathered}$$

Thus, the equation of ellipse will be as follows,

$$\begin{gathered} \frac{{(x-h)}^2}{a^2}+\frac{{(y-k)}^2}{b^2}=1 \\ \frac{(x-{(-1))}^2}{2^2}+\frac{{(y-1)}^2}{1^2}=1 \\ \frac{{(x+1)}^2}{4}+{(y-1)}^2=1 \end{gathered}$$