5. On a circle $\odot O$ with radius 1, take a fixed point $A$ and a moving point $B$. Let point $P$ satisfy $A P / / O B$, and $\overrightarrow{A P} \cdot \overrightarrow{A B}=1$. Then the locus of point $P$ is ().
(A) Ellipse
(B) Parabola
(C) Hyperbola
(D) Any of the above is possible
5. B.

Let's assume $O(0,0)$, $A(1,0)$, and $P(x, y)$.
Since $A P / / O B$, we can set $B(k(x-1), k y)$.
Substituting these coordinates into $\overrightarrow{A P} \cdot \overrightarrow{A B}=1$, we get
$$
k=\frac{x}{(x-1)^{2}+y^{2}} \text {. }
$$

Using the fact that point $B$ lies on $\odot O$, we can derive that the equation satisfied by $(x, y)$ is
$$
x^{2}=(x-1)^{2}+y^{2} \Rightarrow y^{2}=2 x-1 \text {. }
$$

Thus, the locus of point $P$ is a parabola.