5. The number of common points of the curves $\rho=11 \cos \frac{\theta}{2}$ and $\rho=\cos \frac{\theta}{2}$ in the polar coordinate plane is ( ).
A. 0
B. 2
c. 3
D. 4
5. D

When $\rho=0$, the point $(0,0)$ lies on both curves. When $\rho \neq 0$, then from $1+\cos \frac{\theta}{2}=$ $\cos \frac{\theta+2 k \pi}{2} \rightarrow \cos \frac{\theta}{2}=-\frac{1}{2}, \rho=\frac{1}{2}$, which means the points $\left(\frac{1}{2}, \frac{4 \pi}{3}\right)\left(\frac{1}{2},-\frac{4 \pi}{3}\right)$ lie on both curves. From $\mathrm{I}+\cos \frac{\theta}{2}=-\cos \frac{\theta+2 k \pi+\pi}{2}$, we get $\cos \frac{\theta}{2}=0$, which means the point $(1, \pi)$ lies on both curves. Therefore, the answer is $\mathrm{D}$.