Question

In the following exercises, graph each ellipse.

$16x^2+36y^2=576$

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 $16x^2+36y^2=576$

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 

$$\begin{gathered} 16x^2+36y^2=576\text{Given} \\ \frac{16x^2}{576}+\frac{36y^2}{576}=\frac{576}{576}\text{Dividewith576onbothsides} \\ \frac{x^2}{36}+\frac{y^2}{16}=1\text{Simplify} \end{gathered}$$

Step 4.Find the center and major axis

$\frac{x^2}{36}+\frac{y^2}{16}=1$

Compare the given equation with the standard form of the ellipse

$$\begin{gathered} \text{Center:(0,0),} \\ 36>16\text{and} \\ {a}^2=36\Rightarrow a=\pm 6 \\ {b}^2=16\Rightarrow b=\pm 4 \end{gathered}$$

Major axis:

On$x-$axis with endpoints$(-6,0),(6,0).$

Minor axis

On $y-$axis with endpoints$(0,-4),(0,4).$

Step 5. Sketch the ellipse

The graph of the ellipse is shown below