Chapter 8.4, Problem 61E

To determine

To find: Parametric equations for the given curve.

Answer to Problem 61E

Parametric equations are

  $x=a\left(\cot \theta +\sin \theta \cos \theta \right)$

  $y=a\left(1+{\sin }^2\theta \right)$

Explanation of Solution

Given:

  

Calculation:

Create $a$ the first equation for the Longbow curve for the point $P$ . In order to do this find the average of $2a\cot \theta$ which is the length to $R$ and $2a\sin \theta \cos \theta$ is the length to $Q$ to find the midpoint which is located at $P$ .

  $x=\frac{\left(2a\cot \theta +2a\sin \theta \cos \theta \right)}{2}$ Take average of the length

  $x=a\left(\cot \theta +\sin \theta \cos \theta \right)$ Reduce

Create $a$ the second equation for the Longbow curve for the point $P$ . In order to do this find the average of $2a$ which is the height to $R$ and $2a{\sin }^2\theta$ which is the height to $Q$ to find the midpoint which is located at $P$ .

  $x=\frac{2a+2a{\sin }^2\theta }{2}$ Take average of the length

  $y=a\left(1+{\sin }^2\theta \right)$ Reduce

Conclusion: Hence, the parametric equations are

  $x=a\left(\cot \theta +\sin \theta \cos \theta \right)$

  $y=a\left(1+{\sin }^2\theta \right)$