Let S_k = z C : |z - (k + ik)| k for k N , and T = z C : |z - (7 + 7i)| 3 . The number of values of k N such that S_k T is

Step-by-step solution

The set S_k represents a solid disk in the complex plane with center C_k = k + ik and radius R_k = k . The set T represents a solid disk with center C_T = 7 + 7i and radius R_T = 3 . For the two disks to have a non-empty intersection, the distance between their centers must be less than or equal to the sum of their radii: |C_k - C_T| R_k + R_T Calculate the distance between the centers: |C_k - C_T| = |(k-7) + i(k-7)| = (k-7)^2 + (k-7)^2 = 2 |k-7| So the condition becomes: 2 |k-7| k + 3 Case 1: k 7 2 (k-7) k + 3 ( 2