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

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 $d = \sqrt{3^2 + 4^2} = 5$.

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 $\min |z_1 - z_2| = 2$.

$r - (5 + 5) = 2 \Rightarrow r - 10 = 2 \Rightarrow r = 12$

The maximum distance between a point on $C_1$ and a point on $C_2$ is given by $r_1 + d + r_2$.

$\max |z_1 - z_2| = 12 + 5 + 5 = 22$

Answer: $22$