Solution:

Four of the six distances are $a$, so three of the points form an equilateral triangle of side $a$. Call these points $A,B,C$.

The fourth point $D$ is distance $a$ from one of these points, say $A$, and distance $2a$ from another, say $B$. Since $BD$ is a diameter of the circle centered at $A$ with radius $a$, triangle $BCD$ is right.

Thus $b^2=(2a)^2-a^2=3a^2$, so $b/a=\sqrt3$.

Thus, A is the correct answer.