The number of complex numbers z satisfying |z - 3 - i| = |z - 9 -i| and |z - 3 + 3i| = 3 are

Let z = x + iy. Then,

$\begin{array}{l}|\mathrm{z}-3-\mathrm{i}|=|\mathrm{z}-9-\mathrm{i}|\\ \Rightarrow \sqrt{(\mathrm{x}-3{)}^2+(\mathrm{y}-1{)}^2}=\sqrt{(\mathrm{x}-9{)}^2+(\mathrm{y}-1{)}^2}\\ \Rightarrow \mathrm{x}=6\\ |\mathrm{z}-3+3\mathrm{i}|=3\\ \Rightarrow \sqrt{(\mathrm{x}-3{)}^2+(\mathrm{y}+3{)}^2}=3\end{array}$

For x = 6, y = -3.

$\therefore \mathrm{z}=6-3\mathrm{i}$