3. In the Cartesian coordinate system $x O y$, $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{4}=1(a>0)$, respectively. The line $l$ passes through the right vertex $A$ of the hyperbola and the point $B(0,2)$. If the sum of the distances from points $F_{1}$ and $F_{2}$ to the line $l$ equals the length of the segment $B F_{2}$, then the coordinates of point $A$ are $\qquad$
3. $(2 \sqrt{2}, 0)$.

From the given, $c=\sqrt{a^{2}+4}$. It is easy to know that $\sin \angle O A B=\frac{2}{c},\left|B F_{2}\right|=\sqrt{c^{2}+4}$.
The sum of the distances from points $F_{1}$ and $F_{2}$ to line $l$ is
$$
\begin{array}{l}
\left|A F_{1}\right| \sin \angle O A B+\left|A F_{2}\right| \sin \angle O A B \\
=\left|F_{1} F_{2}\right| \frac{2}{c}=4 \\
\Rightarrow \sqrt{c^{2}+4}=4 \Rightarrow c=2 \sqrt{3} \Rightarrow a=2 \sqrt{2} .
\end{array}
$$

Therefore, point $A(2 \sqrt{2}, 0)$.