Let the circles C_1 : |z| = r and C_2 : |z - 3 - 4i| = 5, z C , be such that C_2 lies within C_1. If z_1 moves on C_1, z_2 moves on C_2 and |z_1 - z_2| = 2, then |z_1 - z_2| is equal to:

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Correct answer: C. 22

The center of circle C_1 is O_1(0,0) and its radius is r_1 = r. The center of circle C_2 is O_2(3,4) and its radius is r_2 = 5. The distance between the centers of the two circles is d = 3^2 + 4^2 = 5. Since C_2 lies entirely within C_1, the minimum distance between a point on C_1 and a point on C_2 is given by r_1 - (d + r_2). We are given that |z_1 - z_2| = 2. r - (5 + 5) = 2 r - 10 = 2 r = 12 The maximum distance between a point on C_1 and a point on C_2 is given by r_1 + d + r_2. |z_1 - z_2| = 12 + 5 + 5 = 22 Answer: 22

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