Question

test for symmetry with respect to a. the polar axis, b. the line $\theta=\frac{\pi}{2}$, and c. the pole.
$r^{2}=9\sin 2\theta$

a. is the polar equation symmetrical with respect to the polar axis?
a. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the the polar axis.
b. yes.
c. the polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the polar axis.

b. is the polar equation symmetrical with respect to the line $\theta=\frac{\pi}{2}$?
a. yes.
b. the polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the line $\theta=\frac{\pi}{2}$.
c. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the line $\theta=\frac{\pi}{2}$.

Explanation:

Step1: Test for symmetry about the polar axis ($\theta = 0$)

Replace $\theta$ with $-\theta$.
Original equation: $r^{2}=9\sin(2\theta)$
After replacement: $r^{2}=9\sin(- 2\theta)=-9\sin(2\theta)$ (since $\sin(-x)=-\sin(x)$).
The new equation $r^{2}=-9\sin(2\theta)$ is not the same as the original equation $r^{2}=9\sin(2\theta)$. But failure of the symmetry - test does not guarantee lack of symmetry.

Step2: Test for symmetry about the line $\theta=\frac{\pi}{2}$

Replace $(r,\theta)$ with $(-r,-\theta)$ or $(r,\pi - \theta)$.
Using the replacement $(r,\theta)\to(r,\pi - \theta)$:
Original equation $r^{2}=9\sin(2\theta)$
After replacement: $r^{2}=9\sin(2(\pi - \theta))=9\sin(2\pi - 2\theta)=-9\sin(2\theta)$ (using $\sin(A - B)=\sin A\cos B-\cos A\sin B$ with $A = 2\pi$, $B=2\theta$ and $\sin(2\pi)=0$, $\cos(2\pi)=1$).
The new equation $r^{2}=-9\sin(2\theta)$ is not the same as the original equation $r^{2}=9\sin(2\theta)$. But failure of the symmetry - test does not guarantee lack of symmetry.

Answer:

a. C. The polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the polar axis.
b. B. The polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the line $\theta=\frac{\pi}{2}$.