3. In the complex plane, the points $Z$ that satisfy
$$
|z-1| \sqrt{5}+|z-1-2 \mathrm{i}|=2|z|
$$

form ( ).
(A) a circle
(B) an ellipse
(C) a circular arc
(D) a hyperbola
3.C.

Let $A(0,0), B(1,0), C(1,2)$, and the point corresponding to $z$ be $Z$. From $|z-1| \sqrt{5}+|z-1-2 \mathrm{i}|=2|z|$, we can derive
$$
Z B \cdot A C + Z C \cdot A B = Z A \cdot B C.
$$

By the converse of Ptolemy's theorem, the locus of $Z$ is the arc $\overparen{B C}$ of the circumcircle of $\triangle A B C$ that does not contain point $A$.