Intersection of two circles. Let $C_1$ and $C_2$ be the circles: $\rho=a\sin\theta, \rho=a(\cos\theta + \sin\theta)$ respectively. The graphs of these two circles are 
From the graphs, we see that the intersection points are $(0,0)$, $(\pi/2, a)$. But when we solve the system of equations: $\rho=a\sin\theta, \rho=a(\cos\theta + \sin\theta)$, we obtain $(\theta, \rho)=(\pi/2, a)$ or $(-\pi/2, -a)$. $(\pi/2, a)$, $(-\pi/2, -a)$ are different from $(0,0)$, $(\pi/2, a)$. I am confused. Thank you very much.

The issue here is that one Cartesian point has multiple polar representations.
The two points that you found, $(\pi/2,a)$ and $(-\pi/2,-a)$, both refer to to same location in Cartesian space. Also, you did not find an intersection at the origin because the two circles reach it with different values of $\theta$.
Assuming $a\ne0$, $a\sin\theta=0$ implies that $\theta=0$ or $\theta=\pi$, for $0\le\theta<2\pi$. For the second circle, $a(\cos\theta+\sin\theta)=0$ gives $\theta=3\pi/4$ or $\theta=7\pi/4$ for $0\le\theta<2\pi$. That means that $C_1$ contains the points $(0,0)$ and $(\pi,0)$, while $C_2$ contains the points $(3\pi/4,0)$ and $(7\pi/4,0)$. While these all represent the Cartesian point $(0,0)$, they are not the same in polar form. Whether or not you can consider this to be an intersection depends on the context of the problem.
