Chapter 8.4, Problem 64E

(a)

To determine

To show: the Agnesi curve is written as the given parametric equations

Explanation of Solution

Given:

The parametric equations is

  $\begin{array}{l}x=2a\cot \theta \\ y=2a{\sin }^2\theta \end{array}$

Calculation:

Consider the following sketch of a witch of Agnesi curve. Create a diagram of all the triangles and length using simple trigonometry

  

Create $a$ the first equation for the witch of Agnesi curve for the point $P$ . In order to do these find the $x$ length to the points $P$ using the top large triangle

  $x=2a\cot \theta$ Use large triangle $\cot \theta =\frac{adj}{opp}$

Create the second equation for the witch of Agnesi curve for the point $P$ . In order to do these find the $y$ length to the points $P$ using the top large triangle. See the sketch for the different lengths found and the height will be $2a{\sin }^2\theta$

  $y=2a{\sin }^2\theta$ Solve triangles using simple.

Conclusion:

Hence, the result of the parametric equations for the curve is $x=2a\cot \theta$ and $y=2a{\sin }^2\theta$

(b)

To determine

To graph: the curve using the graphing device.

Answer to Problem 64E

Hence the resultant graph is:

  

Explanation of Solution

Given:

  $a=3$ .

  

Calculation:

Use a graphing device to plot the curve for $a=3,b=2$

  

Conclusion:

Hence the resultant graph is drawn.