Question

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

$4x^2+y^2=4$

Short answer

The equation$4x^2+y^2=4$matches with the graph labeled as I shown below

Step-by-step solution

Step 1. Given Information 

The given equation $4x^2+y^2=4$

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

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,$4x^2+y^2=4$

Simplifying:

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

Since, 4 > 1 therefore it is of the form$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, i.e. vertical ellipse.

Finding values of a,b and c :

$\begin{array}{c}{a}^2=4\\ \Rightarrow a=\pm 2\\ b^2=1\\ \Rightarrow b=\pm 1\\ c=\sqrt{a^2-b^2}\\ \Rightarrow c=\sqrt{4-1}\\ \Rightarrow c=\pm \sqrt{3}\end{array}$

Finding vertices and foci:

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

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