Chapter 9.4, Problem 24E

(a)

To determine

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

Explanation of Solution

Given information: Parametric equations $x=\cos \theta ;y=4\sin \theta$

Calculation:

Parametric equations are given as

  $x=\cos \theta ;y=4\sin \theta$

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

$t$	$x$	$y$
-3	-0.98	-0.56
-2	-0.41	-3.63
-1	0.54	-3.36
0	1	0
1	0.54	3.36
2	-0.41	0.63
3	-0.98	0.56

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 24E

The rectangular equation is $x^2+\frac{y^2}{16}=1$

Explanation of Solution

Given information: Parametric equations $x=\cos \theta ;y=4\sin \theta$

Calculation:

Parametric equations are given as

  $x=\cos \theta ;y=4\sin \theta$

From above equations:

  $\begin{array}{l}x=\cos \theta ---(1)\\ \frac{y}{4}=\sin \theta ---(2)\end{array}$

Adding (1) and (2):

  $\begin{array}{l}{\left(x\right)}^2+{\left(\frac{y}{4}\right)}^2={\cos }^2\theta +{\sin }^2\theta \\ x^2+\frac{y^2}{16}=1\end{array}$

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

Conclusion:

Hence, rectangular equation is $x^2+\frac{y^2}{16}=1$