Chapter 10, Problem 34RE

In Problems 32-34, graph the curve whose parametric equations are given by hand and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility.

$x={\text{sec}}^{2}t$, $y={\text{tan}}^{2}t$; $0\le t\le \frac{\pi }{4}$

To determine

a.

To find: The rectangular equation of the curve.

Answer to Problem 34RE

$x-y=1$ $\{1\le x\le 2\}$

Explanation of Solution

Given:

$x={\text{sec}}^{2}t, y={\text{tan}}^{2}t; 0\le t\le \frac{\pi }{4}$

Calculation:

$x={\text{sec}}^{2}t, y={\text{tan}}^{2}t; 0\le t\le \frac{\pi }{4}$

$1+{\text{tan}}^{2}x={\text{sec}}^{\text{2}}x$

$1+y=x$

$x-y=1$ $\{1\le x\le 2\}$

To determine

b.

To graph: $x={\text{sec}}^{2}t, y={\text{tan}}^{2}t; 0\le t\le \frac{\pi }{4}$, verify with the graph of rectangular equation of the curve.

Answer to Problem 34RE

Explanation of Solution

Given:

$x={\text{sec}}^{2}t, y={\text{tan}}^{2}t; 0\le t\le \frac{\pi }{4}$

Graph:

The graph of $x={\text{sec}}^{2}t, y={\text{tan}}^{2}t; 0\le t\le \frac{\pi }{4}$ is plotted.

The graph of $x-y=1$ $\{1\le x\le 2\}$ is plotted.

Graph 1 represents the graph plotted in parametric mode and the orientation is shown.

The rectangular equation of the curve is calculated. Graph 2 represents the rectangular equation of the curve.

Both the equation represents the same graph and orientation. Hence it is verified.