Chapter 9, Problem 11RE

To determine

To find: The polar equation is $r=4\cos \theta$ .

Answer to Problem 11RE

The polar equation may or may not be symmetric about the line $\theta =\frac{\pi }{2}$ .

Explanation of Solution

Given information:

The given polar equation is $r=4\cos \theta$ .

Calculation:

Calculate the polar equation,

  $\begin{array}{c}4\cos \left(-\theta \right)=4\cos \theta \\ =r\end{array}$

This shows that the function is symmetric about the polar axis.

Now, check the symmetry about the origin.

Therefore,

  $\left(-r\right)=4\cos \theta$

Therefore, this test fails.

Therefore, the function may or may not be symmetric about the origin.

For this let us replace $\theta$ by $\left(\pi -\theta \right)$ .

Therefore, $4\cos \left(\pi -\theta \right)=-4\cos \theta$

Therefore, this test fails.

Therefore, the polar equation may or may not be symmetric about the line $\theta =\frac{\pi }{2}$ .

The graph of this function is as follows figure-(1) shows.

  

Figure-(1)

Therefore, the polar equation may or may not be symmetric about the line $\theta =\frac{\pi }{2}$ .