3. In the Cartesian coordinate system, points whose both coordinates are rational numbers are called rational points. 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
(C)
3.【Analysis and Solution】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 $x=a$, 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 $k=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{y_{1}}{x_{1}-a}$, hence $y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left(x_{1}-a\right)$.
Since $a$ is an irrational number, the sufficient condition for the above equation to hold is $y_{2}=y_{1}$, so the line passing through the point $(a, 0)$ and through two rational points is unique, and its equation is $y=0$.