3. In the Cartesian coordinate system, points with both coordinates being 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
Obviously, the $x$-axis satisfies the condition. If there exists another line passing through two rational points $(m, n),(s, t)(n, t \neq 0)$, then $\frac{n-t}{m-s}=\frac{n}{m-a} \Rightarrow m-a=\frac{n(m-s)}{n-t} \Rightarrow a=m-\frac{n(m-s)}{n-t}=\frac{n s-m t}{n-t}$ is a rational number, which is a contradiction. Therefore, there is and only one line that passes through at least two rational points, so the answer is $C$.