Answer
a. symmetric with respect to the polar axis.
b. symmetric with respect to the line $\theta=\frac{\pi}{2}$.
c. symmetric with respect to the pole.
Work Step by Step
a. To test the symmetry with respect to the polar axis, replace $(r,\theta)$ with $(r,-\theta)$; we have $r^2=9cos(-2\theta)$ or $r^2=9cos(2\theta)$. 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 $(-r)^2=9cos(-2\theta)$ or $r^2=9cos(2\theta)$. Thus the equation is 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 $(-r)^2=9cos(2\theta)$ or $r^2=9cos(2\theta)$. Thus the equation is symmetric with respect to the pole.
