Question

test for symmetry with respect to a. the polar axis, b. the line $\theta=\frac{\pi}{2}$, and c. the pole.
$r^{2}=9\sin 2\theta$
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 $\theta=\frac{\pi}{2}$.
c. is the polar equation symmetrical with respect to the pole?
a. the polar equation failed the test for symmetry which means that the graph may or may not be symmetric with respect to the pole.
b. yes.
c. the polar equation failed the test for symmetry which means that the graph is not symmetric with respect to the pole.

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 is not the same as the original equation. But failure of the symmetry test does not necessarily mean the graph is not symmetric.

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

Replace $(r,\theta)$ with $(-r,-\theta)$.
Original equation: $r^{2}=9\sin(2\theta)$
After replacement: $(-r)^{2}=9\sin(-2\theta)\Rightarrow r^{2}=-9\sin(2\theta)$.
The new equation is not the same as the original equation. Again, failure of the symmetry test does not necessarily mean the graph is not symmetric.

Step3: Test for symmetry about the pole ($r = 0$)

Replace $r$ with $-r$.
Original equation: $r^{2}=9\sin(2\theta)$
After replacement: $(-r)^{2}=9\sin(2\theta)\Rightarrow r^{2}=9\sin(2\theta)$.
The new equation is the same as the original equation.

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 $\theta=\frac{\pi}{2}$.
c. B. Yes.