Chapter 9.4, Problem 23E

(a)

To determine

To calculate: To sketch the curve represented by the parametric equations

Explanation of Solution

Given information: Parametric equations $x=2\cos \theta ;y=3\sin \theta$

Calculation:

Parametric equations are given as

  $x=2\cos \theta ;y=3\sin \theta$

Calculating the values of $x$ and $y$ at various values of $t$

$t$	$x$	$y$
-3	-1.97	-0.42
-2	-0.83	-2.72
-1	1.08	-2.52
0	2	0
1	1.08	2.52
2	-0.83	2.72
3	-1.97	0.42

Graph of the function is as follows:

The arrows on the curve indicates its orientation as $t$ increases from $-3$ to $3$

  

Conclusion:

Hence, graph is sketched

(b)

To determine

To calculate: To write the rectangular equations for the parametric equations

Answer to Problem 23E

The rectangular equation is $\frac{x^2}{4}+\frac{y^2}{9}=1$

Explanation of Solution

Given information: Parametric equations $x=2\cos \theta ;y=3\sin \theta$

Calculation:

Parametric equations are given as

  $x=2\cos \theta ;y=3\sin \theta$

From above equations:

  $\begin{array}{l}\frac{x}{2}=\cos \theta ---(1)\\ \frac{y}{3}=\sin \theta ---(2)\end{array}$

Adding (1) and (2):

  $\begin{array}{l}{\left(\frac{x}{2}\right)}^2+{\left(\frac{y}{3}\right)}^2={\cos }^2\theta +{\sin }^2\theta \\ \frac{x^2}{4}+\frac{y^2}{9}=1\end{array}$

Here, domain is $0<\theta <2\pi$

Conclusion:

Hence, rectangular equation is $\frac{x^2}{4}+\frac{y^2}{9}=1$