Maximum Distance Between Points on Concentric Like Circles

To find the maximum distance $\max |z_1 - z_2|$ between points $z_1 \in C_1$ and $z_2 \in C_2$, we analyze the geometry of the two circles in the complex plane.

1. Identify the Centers and Radii

• For circle $C_1: |z| = r$, the center is $O = 0 + 0i$ and the radius is $r_1 = r$.
• For circle $C_2: |z - (3 + 4i)| = 5$, the center is $A = 3 + 4i$ and the radius is $r_2 = 5$.

2. Distance Between Centers

The distance $d$ between the centers $O$ and $A$ is given by: $d = |A - O| = |3 + 4i| = \sqrt{3^2 + 4^2} = 5$

3. Determine the Radius $r$

Since circle $C_2$ lies completely inside circle $C_1$, the minimum distance between a point $z_1$ on $C_1$ and a point $z_2$ on $C_2$ occurs along the line segment connecting their centers $O$ and $A$: $\min |z_1 - z_2| = r_1 - (d + r_2)$

We are given that $\min |z_1 - z_2| = 2$. Substituting the known values into the equation: $2 = r - (5 + 5)$ $2 = r - 10 \implies r = 12$

4. Find the Maximum Distance

The maximum distance between a point $z_1$ on $C_1$ and a point $z_2$ on $C_2$ occurs when $z_1$ and $z_2$ lie on opposite ends along the line passing through centers $O$ and $A$: $\max |z_1 - z_2| = r_1 + d + r_2$

Substituting $r_1 = 12$, $d = 5$, and $r_2 = 5$: $\max |z_1 - z_2| = 12 + 5 + 5 = 22$

Thus, the correct option is C.