Chapter 10.7, Problem 22AYU

In Problems 7-26, graph the curve whose parametric equations are given and show its orientation. Find the rectangular equation of each curve. Verify your graph using a graphing utility.

$x=2\cos t$, $y=\sin t$; $0\le t\le \frac{\pi }{2}$

To determine

a.

To find: The rectangular equation of the curve.

Answer to Problem 22AYU

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

Explanation of Solution

Given:

$x=2\cos t,y=\sin t;\text{ 0 }\le t\le \frac{\pi }{2}$

Calculation:

$x=2\cos t,y=\sin t;\text{ 0 }\le t\le \frac{\pi }{2}$

$x^2/4=\left(1-{\sin }^{2}t\right)$

$\frac{x^2}{4}=\left(1-y^2\right)$

$\frac{x^2}{4}+y^2=1$; $x,y\ge 0$

To determine

b.

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

Answer to Problem 22AYU

Explanation of Solution

Given:

$x=2\cos t,y=\sin t;\text{ 0 }\le t\le \frac{\pi }{2}$

Graph:

The graph of $x=2\cos t,y=\sin t$ is plotted.

The graph of $\frac{x^2}{4}+y^2=1$; $y\ge 0$ is plotted.

Graph 1 represents the graph of $x=2\cos t,y=\sin t;\text{ 0 }\le t\le \frac{\pi }{2}$ plotted in parametric mode and the orientation is shown.

The rectangular equation of the curve is calculated. Graph 2 represents the curve of the rectangular equation $\frac{x^2}{4}+y^2=1$; $y\ge 0$. Both the equation represents the same graph and orientation. Hence it is verified.