Đặt \({\rm{w}} = x + yi,\) ta có:

\({\rm{w}} = \left( {1 + i} \right)z + 1 \Leftrightarrow {\rm{w}} = \left( {1 + i} \right)\left( {z - 1} \right) + i + 2 \Leftrightarrow w - i - 2 = \left( {z - 1} \right) + i\left( {z - 1} \right)\)

\( \Leftrightarrow \left| {w - i - 2} \right| = \left| {\left( {z - 1} \right) + i\left( {z - 1} \right)} \right| \Leftrightarrow {\left( {x - 2} \right)^2} + {\left( {y - 1} \right)^2} = 2{\left( {z - 1} \right)^2} \le 2 \Rightarrow R = \sqrt 2 \)

\( \Rightarrow S = \pi {R^2} = 2\pi .\)