91Ó°ÊÓ

We have to match the equation with its graph.

We know that

The equation of an ellipse with its center in the point (0,0) is given with:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

if the major axis is parallel to the x-axis, or with

$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$

if the major axis is parallel to the y-axis.

In both cases$a>b$

In this case, from the graph we can see that the major axis is parallel to x-axis since its passes through the vertices which are located on the x-axis.

From the previous step we get that the ellipse has to have an equation of the following form:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

From the graph we can see that the vertices are in the following points $(2,0)and(-2,0)$.

From the fact that any ellipse that has its major axis parallel to the x-axis, has its vertices located in$(-a,0)and(a,0)$

it follows that $a=2$.

Since the height of the ellipse is visible to be $b=1$it follows that the equation of the ellipse is

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

The correct answer is$\left(A\right)\frac{x^2}{4}+\frac{y^2}{1}=1$