Final answer:
The number of possible integral values of r when two circles have common points is infinite. Option D is correct.
Step-by-step explanation:
If the circles A and B have common points, then the number of possible integral values of r is infinite (d).
When two circles intersect, they have two common points. The circles can be represented by the equations (x-a)^2 + (y-b)^2 = r^2, where (a,b) are the coordinates of the center and r is the radius of the circle. If the circles have common points, it means that the distance between their centers is less than or equal to the sum of their radii.
To find the number of possible values of r, we need to consider the range of distances between the centers of the circles. Let d be the distance between the centers of the circles and R1, R2 be the radii of circles A and B, respectively. Then R1 + R2 >= d and R1 - R2 <= d. Since R1 and R2 are positive, the range of possible values for d is R1 - R2 <= d <= R1 + R2.
For any integral value of d within this range, there will be a corresponding integral value of r such that the circles have common points. Therefore, the number of possible integral values of r is infinite.