Answer
a. symmetric with respect to the polar axis.
b. may not be symmetric with respect to the line $\theta=\frac{\pi}{2}$.
c. may not be symmetric with respect to the pole.
Work Step by Step
a. Given $r=5+3cos\theta$. To test the symmetry with respect to the polar axis, replace $(r,\theta)$ with $(r,-\theta)$. We have $r=5+3cos(-\theta)$, which gives $r=5+3cos\theta$. Thus the equation is symmetric with respect to the polar axis.
b. To test the symmetry with respect to the line $\theta=\frac{\pi}{2}$, replace $(r,\theta)$ with $(-r,-\theta)$. We have $-r=5+3cos(-\theta)$, which gives $r=-5-3cos\theta$. Thus the equation may not be symmetric with respect to the line $\theta=\frac{\pi}{2}$.
c. To test the symmetry with respect to the pole, replace $(r,\theta)$ with $(-r,\theta)$. We have $-r=5+3cos(\theta)$, which gives $r=-5-3cos\theta$. Thus the equation may not be symmetric with respect to the pole.
