Circle Properties

We are given two circles in the complex plane:

• Circle $C_1$: $|z| = r$. This circle is centered at the origin $O=0$ with radius $R_1 = r$.
• Circle $C_2$: $|z - (3 + 4i)| = 5$. This circle is centered at $c_2 = 3 + 4i$ with radius $R_2 = 5$.

The problem states that $C_2$ lies entirely within $C_1$. This geometric condition implies $R_1 > d + R_2$, where $d$ is the distance between the centers.

Distance Between Centers

Calculate the distance $d$ between the centers of $C_1$ (origin $0$) and $C_2$ ($3+4i$):

$ d = |0 - (3 + 4i)| = |-(3 + 4i)| = \sqrt{(-3)^2 + (-4)^2} $

$ d = \sqrt{9 + 16} = \sqrt{25} = 5 $

Minimum Distance Calculation

When one circle ($C_2$) is inside another ($C_1$), the minimum distance between points $z_1$ on $C_1$ and $z_2$ on $C_2$ is given by the difference between the radii adjusted by the distance between centers:

$ \min|z_1 - z_2| = R_1 - (d + R_2) $

We are given $\min|z_1 - z_2| = 2$. Substitute the known values:

$ 2 = r - (5 + 5) $

$ 2 = r - 10 $

Solving for the radius $r$ of $C_1$:

$ r = 2 + 10 = 12 $

The radius of $C_1$ is $R_1 = 12$. Note that $12 > 5 + 5$, consistent with $C_2$ being inside $C_1$.

Maximum Distance Calculation

The maximum distance between points $z_1$ on $C_1$ and $z_2$ on $C_2$ occurs along the line connecting their centers, extending outwards:

$ \max|z_1 - z_2| = R_1 + d + R_2 $

Substitute the values $R_1 = 12$, $d = 5$, and $R_2 = 5$:

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

$ \max|z_1 - z_2| = 22 $