Question

In the following exercises, graph each ellipse.

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

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

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

Step 2. Concept used

The standard form of the ellipse with

center $(0,0)$is

localid="1645422226494" $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

• $a>b\Rightarrow$major axis on$x-$axis with x-intercepts$\left(-a,0\right),(a,0).$
• $b>a\Rightarrow$major axis on $y-$axis with $y-$intercepts$\left(0,-b\right),(0,b).$

Step 3. Find the center and major axis

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

Compare with the standard form of the ellipse

Center$\left(0,0\right).$

Here,$9>4$

$$\begin{gathered} a^2=4\Rightarrow a=\pm 2 \\ {b}^2=9\Rightarrow b=\pm 3 \end{gathered}$$

So,

$y-$axis is the major axis with endpoints localid="1645424530502" $\left(0,-3\right)\text{,(0,3).}$

and

$x-$axis is the minor axis with endpoints$\left(-2,0\right),(2,0).$

Step 4. Graph the ellipse

The graph of the ellipse is shown below.