Question

Graphs of Ellipses Match the equation with the graphs labeled I-IV. Given reasons for your answers.

$x^2+\frac{y^2}{9}=1$

Short answer

The equation $x^2+\frac{y^2}{9}=1$ matches with the graph labeled IV as shown below

Step-by-step solution

step 1. Given Information

The equation of an ellipse,$x^2+\frac{y^2}{9}=1$

Step 2. The matching graph of the ellipse among the graphs labeled I-IV

We will identify the values of a and b by comparing with $x^2+\frac{y^2}{9}=1$

Then by comparing the values of a and b, we will decide whether it is a horizontal or vertical ellipse.

The value ofc is given as

$c=\sqrt{a^2-b^2}$

Finally, with the vertices and foci of the ellipse, we will match the correct graph of the ellipse.

Step 3. Identifying Horizontal or Vertical ellipse:

Given ellipse,$x^2+\frac{y^2}{9}=1$

Since 9 > 1 therefore it is of the form$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$, i.e. vertical ellipse.

Finding values of a, b andc :

$\begin{array}{c}{a}^2=9\\ \Rightarrow a=\pm 3\\ b^2=1\\ \Rightarrow b=\pm 1\\ c=\sqrt{a^2-b^2}\\ \Rightarrow c=\sqrt{9-1}\\ \Rightarrow c=\pm 2\sqrt{2}\end{array}$

Finding vertices and foci:

$\begin{array}{l}(0,\pm a)\equiv (0,\pm 3)\\ (0,\pm c)\equiv (0,\pm 2\sqrt{2})\end{array}$

Analyzing these values with the four graphs,we find that graph IV matches as shown below: