Chapter 9.4, Problem 29E

To determine

To Graph:

The polar equation of $r=4\sec \left(\theta -\frac{\pi }{4}\right)$ .

Explanation of Solution

Given information:

The given function is $r=4\sec \left(\theta -\frac{\pi }{4}\right)$ .

Consider the polar equation of $r=4\sec \left(\theta -\frac{\pi }{4}\right)$ ,

  $\begin{array}{l}r=4\sec \left(\theta -\frac{\pi }{4}\right)\\ r=4\cdot \frac{1}{\cos \left(\theta -\frac{\pi }{4}\right)}\\ r\cos \left(\theta -\frac{\pi }{4}\right)=4\\ 4=r\cos \theta \cos \frac{\pi }{4}+r\sin \theta \sin \frac{\pi }{4}\\ 4=r\cos \theta \times \frac{1}{\sqrt{2}}+r\sin \theta \times \frac{1}{\sqrt{2}}\end{array}$

Let $x=r\cos \theta$ and $y=r\sin \theta$

  $1=\frac{1}{4\sqrt{2}}\left(x+y\right)$

For $\theta =0$ we have $r=\frac{4}{\cos \left(-\frac{\pi }{4}\right)}=\frac{4}{\frac{1}{\sqrt{2}}}=4\sqrt{2}\Rightarrow \left(4\sqrt{2},0\right)$

For $\theta =\frac{\pi }{4}$ we have $r=\frac{4}{\cos \left(0\right)}=\frac{4}{1}=4\Rightarrow \left(4,\frac{\pi }{4}\right)$

Graph: