Chapter 11.2, Problem 49E

a.

To determine

To graph the top half of the ellipses using graphing device.

Explanation of Solution

Given information: Use graphing device to graph the ellipse ${\text{x}}^2{\text{+ky}}^2=100$ where $\text{k}=4,10,25\text{ and 50}$ .

Calculation:

For $\text{k}=4$ , the equation of the ellipse can be calculated by:

  $\begin{array}{l}{\text{x}}^2{\text{+ky}}^2=100\\ {\text{x}}^2+4{\text{y}}^2=1\\ \frac{{\text{x}}^2}{100}+\frac{{\text{y}}^2}{25}=1\\ \text{y}=\pm 5\sqrt{\left(1-\frac{{\text{x}}^2}{100}\right)}\end{array}$

For $\text{k}=10$ , the equation of the ellipse can be calculated by:

  $\begin{array}{l}{\text{x}}^2{\text{+ky}}^2=100\\ {\text{x}}^2+10{\text{y}}^2=1\\ \frac{{\text{x}}^2}{100}+\frac{{\text{y}}^2}{10}=1\\ \text{y}=\pm \sqrt{10\left(1-\frac{{\text{x}}^2}{100}\right)}\end{array}$

For $\text{k}=25$ , the equation of the ellipse can be calculated by:

  $\begin{array}{l}{\text{x}}^2{\text{+ky}}^2=100\\ {\text{x}}^2+25{\text{y}}^2=1\\ \frac{{\text{x}}^2}{100}+\frac{{\text{y}}^2}{4}=1\\ \text{y}=\pm 2\sqrt{\left(1-\frac{{\text{x}}^2}{100}\right)}\end{array}$

For $\text{k}=50$ , the equation of the ellipse can be calculated by:

  $\begin{array}{l}{\text{x}}^2{\text{+ky}}^2=100\\ {\text{x}}^2+50{\text{y}}^2=1\\ \frac{{\text{x}}^2}{100}+\frac{{\text{y}}^2}{2}=1\\ \text{y}=\pm \sqrt{2\left(1-\frac{{\text{x}}^2}{100}\right)}\end{array}$

Since, only the top part of the graph is to be made, only the positive part of the equation is to be used.

Graph: Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expressions

  $\text{y}=5\sqrt{\left(1-\frac{{\text{x}}^2}{100}\right)},\text{y}=\sqrt{10\left(1-\frac{{\text{x}}^2}{100}\right)},\text{y}=2\sqrt{\left(1-\frac{{\text{x}}^2}{100}\right)}\text{ and y}=\sqrt{2\left(1-\frac{{\text{x}}^2}{100}\right)}$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

b.

To determine

To determine the similarities and differences in the family of ellipses drawn.

Answer to Problem 49E

The family of ellipses have the same length of major axis but different lengths of minor axis.

Explanation of Solution

Given information:

The following graph for the family of ellipses is drawn:

  

From the above graph it is seen that the family of ellipses have the same length of major axis but different lengths of minor axis.