Question

In the following exercises, graph each ellipse.

$16x^2+9y^2=144$

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 $16x^2+9y^2=144$

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

$$\begin{gathered} 16x^2+9y^2=144\text{Given} \\ \frac{16x^2}{144}+\frac{9y^2}{144}=\frac{144}{144}\text{Dividewith}144\text{onbothsides} \\ \frac{x^2}{9}+\frac{y^2}{16}=1\text{Simplify} \end{gathered}$$

Compare the given equation with the standard form of the ellipse

$$\begin{gathered} \text{Center:(0,0),} \\ 16>9\text{and} \\ {a}^2=9\Rightarrow a=\pm 3 \\ {b}^2=16\Rightarrow b=\pm 4 \end{gathered}$$

Major axis:

on y-axis with end oints $(0,-4),(0,4).$

Minor axis:

on role="math" localid="1645448767451" $x-$axis with endpoints$(-3,0),(3,0).$

Step 4. Sketch the graph

The graph of the ellipse is shown below