Answer
a. symmetric with respect to the polar axis.
b. may or may not be symmetric with respect to the line $\theta=\frac{\pi}{2}$.
c. may or may not be symmetric with respect to the pole.
Work Step by Step
a. We are given the polar equation $r=1-4cos\theta$. Tto test the symmetry with respect to the polar axis, let $\theta\to -\theta$; we have $r=1-4cos(-\theta)=1-4cos\theta$. Thus, the equation is symmetric with respect to the polar axis.
b. To test the symmetry with respect to the line $\theta=\frac{\pi}{2}$, let $r\to -r$ and $\theta\to -\theta$; we have $-r=1-4cos(-\theta)=1-4cos\theta$ or $r=-1+4cos\theta$. Thus, the equation may or may not be symmetric with respect to the line $\theta=\frac{\pi}{2}$.
c. To test the symmetry with respect to the pole, let $r\to -r$; we have $-r=1-4cos\theta$ or $r=-1+4cos\theta$. Thus, the equation may or may not be symmetric with respect to the pole.
