Question

In the following exercises, graph each ellipse.

$4x^2+25y^2=100$

Short answer

The graph of the ellipse is

Step-by-step solution

Step 1.Given information

The equation of the ellipse is$4x^2+25y^2=100$

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. Convert into standard form

$$\begin{gathered} 4x^2+25y^2=100\text{Given} \\ \frac{4x^2}{100}+\frac{25y^2}{100}=\frac{100}{100}\text{Dividewith}100\text{onbothsides} \\ \frac{x^2}{25}+\frac{y^2}{4}=1\text{Simplify} \end{gathered}$$

Step 4. Find the center and major axis

Compare the given equation with the standard form of the ellipse

$\frac{x^2}{25}+\frac{y^2}{4}=1$

Center is $(0,0)$

$25>4$

$$\begin{gathered} a^2=25\Rightarrow a=\pm 5 \\ {b}^2=4\Rightarrow b=\pm 2 \end{gathered}$$

The major axis is on $x-$axis with endpoints$(-5,0),(5,0).$

The minor axis is on$y-$axis with endpoints$(0,-2),(0,2).$

Step 5. Sketch the graph

The graph of the ellipse is