A & B are points that given
Can I draw more than two circle (with a given radius) at A & B? [The circles are with the same radius]
In plane geometry (i.e. in 2-dimensions) there are either zero or two circles of a given radius R that can contain two given points (the zero case comes about if the points are separated by a distance greater than 2R). In 3-dimensional space two points don't define a plane, and hence there may be an infinite number of circles of a given radius that include the two points, all aligned on different planes, or again there may be zero such circles if the points are too far apart.
Good point, my bad. I was thinking of cases where 2R is less than or greater than the distance between the points A and B. For example if R = sqrt(2) then the center of a circle passing through (0,0) and (2,0) coud be either at (1, 1) or (1, -1). I should have thought about the case where 2R exactly equals the distance between A and B, which yields only one circle, in this case with its center at (1,0).
It surely clear that this question is about a plane (2-D). So lets say that $A~\&~B$ are two points.
Then $d=D(A,B)>0$ the distance between the points is positive.
Now I agree with Dennis the could have been a clearer posting. I am not clear about the given $R$ radius.
Suppose that $\ell$ is the perpendicular bisector of the line segement $\overline{AB}$. Now any circle of radius $R$ that contains $A~\&~B$ has its center on $\ell$ If $R=0.5d$ there is only one center on $\ell$ If $R<0.5d$ there are no centers. If $R>0.5d$ then there are exactly two centers on $\ell$