2. For the arithmetic sequence $\left.\mid a_{n}\right\}$, the first term $a_{1}=8$, and there exists a unique $k$ such that the point $\left(k, a_{k}\right)$ lies on the circle $x^{2}+y^{2}=10^{2}$. Then the number of such arithmetic sequences is $\qquad$.
2.17.

Obviously, the point $(1,8)$ is inside the circle, and the points on the circle $(1, \pm \sqrt{9 a})$

do not satisfy the general term formula of the arithmetic sequence
$$
a_{k}=a_{1}+(k-1) d .
$$

However, the points $\left(k, \pm \sqrt{100-k^{2}}\right)(k=2,3, \cdots, 10)$ on the circle
$$
k^{2}+a_{k}^{2}=100
$$

can satisfy the arithmetic sequence (1), provided that the common difference is
$$
d=\frac{ \pm \sqrt{100-k^{2}}-8}{k-1} .
$$

When $k=2,3, \cdots, 10$, 17 common differences can be derived, corresponding to 17 arithmetic sequences with the same first term.

Note: Here, the line (1) intersects the circle (2) at one point. If we solve the system (1) and (2) and calculate the discriminant to be 0, it will yield no solution, because the intersection point is not a tangent point.