2. In $\triangle A B C$, $A B=2$ is a fixed length. From $A$, draw the median $A D$ to $B C$, $A D=1 \frac{1}{2}$. Also, point $D$ is a moving point, then the locus of vertex $C$ of the triangle is
A. The line $A B$, from $A$ with a length of $1 \frac{1}{2}$
B. A circle, centered at $A$, with a radius of 2
C. A circle, centered at $A$, with a radius of 3
D. A circle, with a radius of 3, and its center on $A B$, 4 units from $B$
E. An ellipse, with $A$ as one of the foci
2. D Taking $A$ as the origin and the side $A B$ as the x-axis to establish a coordinate system, the coordinates of $A$ and $B$ are $(0,0)$ and $(2,0)$, respectively. Let the coordinates of $C$ be $(x, y)$, then the coordinates of the midpoint $D$ of $B C$ are $\left(\frac{x+2}{2}, \frac{y}{2}\right)$. By $A D=\sqrt{\left(\frac{x+2}{2}-0\right)^{2}+\left(\frac{y}{2}-0\right)^{2}}=\frac{3}{2}$, we get $(x+2)^{2}+y^{2}=9$, which is a circle with center at $(-2,0)$ and radius 3.