Let, $$z = x + iy$$

Given, $$1 \le \left| {z - (1 + i)} \right| \le 2$$

$$ \Rightarrow 1 \le \left| {x + iy - 1 - i} \right| \le 2$$

$$ \Rightarrow 1 \le \left| {(x - 1) + i(y - 1)} \right| \le 2$$

$$ \Rightarrow 1 \le \sqrt {{{(x - 1)}^2} + {{(y - 1)}^2}} \le 2$$

It represent two concentric circle both have center at (1, 1) and radius 1 and 2.

Also given,

$$\left| {z - (1 - i)} \right| = 1$$

$$ \Rightarrow \left| {x + iy - 1 + i} \right| = 1$$

$$ \Rightarrow \left| {(x - 1) + i(y + 1)} \right| = 1$$

$$ \Rightarrow \sqrt {{{(x - 1)}^2} + {{(y + 1)}^2}} = 1$$

This represent a circle with center at (1, $$-$$1) and radius = 1.

In the common region infinite values of B possible.