3. (C)

If the line $l$ passes through the point $(a, 0)$ and through two rational points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$, then it must be that $x_{1} \neq x_{2}$.

Otherwise, the equation of $l$ would be $x=a$, and thus $x_{1}=x_{2}=a$, which contradicts the fact that $\left(x_{1}, y_{1}\right)$, $\left(x_{2}, y_{2}\right)$ are rational points.
The slope of the line $l$ is
$$
k=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{y_{1}}{x_{1}-a},
$$

thus
$$
y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left(x_{1}-a\right) .
$$

Since $a$ is an irrational number, the above equation holds if and only if $y_{1}=y_{2}$, hence the line passing through the point $(a, 0)$ and through two rational points is unique, and its equation is $y=0$.
3. In the Cartesian coordinate system, a point whose both coordinates are rational numbers is called a rational point. If $a$ is an irrational number, then among all lines passing through the point $(a, 0)$,
(A) there are infinitely many lines, each of which contains at least two rational points;
(B) there are exactly $n(2 \leqslant n<+\infty)$ lines, each of which contains at least two rational points;
(C) there is exactly one line that passes through at least two rational points;
(D) each line passes through at most one rational point.