Chapter 9.5, Problem 55E

To determine

To calculate: To convert the rectangular equation to polar form

Answer to Problem 55E

The polar equation is $r^2=8\csc 2\theta$

Explanation of Solution

Given information: Rectangular equation is $xy=4$

Formula Used:

Polar coordinate of a point $\left(r,\theta \right)$ in rectangular form is given as

  $\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$

Calculation:

Rectangular equation is given as

  $xy=4$

Substituting $x=r\cos \theta ;y=r\sin \theta$ in above equation:

  $\begin{array}{l}\left(r\cos \theta \right)\left(r\sin \theta \right)=4\\ r^2\sin \theta \cos \theta =4\\ 2\times r^2\sin \theta \cos \theta =2\times 4\\ r^2\left(2\sin \theta \cos \theta \right)=8\\ r^2\sin 2\theta =8\\ r^2=\frac{8}{\sin 2\theta }\\ r^2=8\csc 2\theta \end{array}$

Thus, polar equation is $r^2=8\csc 2\theta$

Conclusion:

Hence, polar equation is $r^2=8\csc 2\theta$