Question 11.3.2: Graphing an Ellipse Sketch a graph of the ellipse whose equa...

First, write the equation in standard form.

\begin{aligned}&9 x^{2}+4 y^{2}=36 \quad \text { Given equation }\\&\frac{9 x^{2}}{36}+\frac{4 y^{2}}{36}=1 \quad \text { Divide both sides by } 36\\&\underset{\overbrace{\text{larger denominator}}^{} }{\frac{x^{2}}{4}+\frac{y^{2}}{9}=1} \quad \text { Simplify. }\end{aligned}

Because the denominator in the y²-term is larger than the denominator in the x²-term, the ellipse is a vertical ellipse. Here a^{2}=9 \text { and } b^{2}=4 \text {, so } c^{2}=a^{2}-b^{2}=9-4=5. Thus, a=3, b=2, \text { and } c=\sqrt{5}. From the table on the previous page, we know the following features of the vertical ellipse:

\begin{array}{lrl}\text { Vertices: } & (0, \pm a) & =(0, \pm 3) \\\text { Foci: } & (0, \pm c) & =(0, \pm \sqrt{5}) \\\text { Length of major axis: } & 2 a & =2(3)=6 \\\text { Length of minor axis: } & 2 b & =2(2)=4\end{array}

The graph of the ellipse is shown in Figure 16.