Let A = z C : 4 |z| 7 and B = z C : |z - 2| = r , where r > 0 . If A B contains exactly one element, then the product of all possible values of r is

Step-by-step solution

The set A represents an annulus centered at the origin with inner radius 4 and outer radius 7 . The set B represents a circle centered at z₀ = 2 with radius r . For the intersection A B to contain exactly one element, the circle B must be tangent to one of the boundary circles of A and lie entirely outside the interior of the annulus. The distance from the origin to the center of B is |2 - 0| = 2 . The distances from the origin to points on B range from |r - 2| to r + 2 . For exactly one point of intersection, the