Question

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

$16x^2+25y^2=400$

Short answer

The equation $16x^2+25y^2=400$matches with the graph labeledIII as shown below:

Step-by-step solution

Step 1. Given Information

The equation of an ellipse, $16x^2+25y^2=400$

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

We will identify the values ofaandbby 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 of c 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.

Identifying Horizontal or Vertical ellipse:
Given ellipse, $16x^2+25y^2=400$
Simplifying

$\frac{x^2}{25}+\frac{y^2}{16}=1$

Since ,25>16 therefore it is of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, i.e. horizontal ellipse.

Finding values of a ,b and c :

$\begin{array}{c}{a}^2=25\\ \Rightarrow a=\pm 5\\ b^2=16\\ \Rightarrow b=\pm 4\\ c=\sqrt{a^2-b^2}\\ \Rightarrow c=\sqrt{25-16}\\ \Rightarrow c=\pm 3\end{array}$

Finding vertices and foci:

$(\pm a,0)\equiv (\pm 5,0)$

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