Question

In the following exercises, graph each ellipse.

$x^2+\frac{y^2}{4}=1$

Short answer

The Graph of the given ellipse is given by

Step-by-step solution

Step 1.Given information

The equation of the ellipse is

$x^2+\frac{y^2}{4}=1$

Step 2.  Concept used

The standard form of the ellipse

with Center at the origin is

$$\begin{gathered} \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \\ a>b\Rightarrow \text{majoraxisis}x-\text{axiswithendpoints}\left(-a,0\right)\text{,(a,0)} \\ b>a\Rightarrow \text{majoraxisis}y-\text{axis.withendpoints}\left((0,-b\right)\text{,(0,b)} \end{gathered}$$

Step 3.Find the center and major axis

$$\begin{gathered} x^2+\frac{y^2}{4}=1\text{Given} \\ \frac{x^2}{1}+\frac{y^2}{4}=1\text{Rewritetheequation} \end{gathered}$$

Compare the given equation with the standard form of the ellipse

$$\begin{gathered} \text{Center:(0,0),} \\ 4>1\text{and} \\ {a}^2=1\Rightarrow a=\pm 1 \\ {b}^2=4\Rightarrow b=\pm 2 \end{gathered}$$

Major axis:

On $x-$axis with endpoints$(-1,0),(1,0).$

Minor axis

On $y-$axis with endpoints$\left(0,2),(0,-2\right)$

Step 4. Sketch the graph

The graph of the ellipse is shown below