The equation $\mathrm{z}\overline{\mathrm{z}}+(2-3\mathrm{i})\mathrm{z}+(2+3\mathrm{i})\overline{\mathrm{z}}+4=0$ represents a circle of radius

$$\begin{gathered} \mathrm{z} \overline{\mathrm{z}}+(2-3\mathrm{i})\mathrm{z}+(2+3\mathrm{i}) \overline{\mathrm{z}}+4=0 \\ \Rightarrow (\mathrm{x}+\mathrm{iy}) (\mathrm{x}-\mathrm{iy})+(2-3\mathrm{i}) (\mathrm{x}+\mathrm{iy})+(2+3\mathrm{i}) (\mathrm{x}-\mathrm{iy})+4=0 \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{x}+3\mathrm{y}+2\mathrm{x}+3\mathrm{y}+4=0 \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2+4\mathrm{x}+6\mathrm{y}+4=0 \\ \therefore \mathrm{r}=\sqrt{4+9-4}=3 \end{gathered}$$

$$\begin{gathered} \boxed{\mathbf{\text{General equation of circle is}} \\ \mathit{z}\overline{\mathbf{z}}\mathbf{+}\mathit{a}\overline{\mathbf{z}}\mathbf{+}\overline{\mathbf{a}}\mathit{z}\mathbf{+}\mathit{b}\mathbf{=}\mathbf{0}\mathbf{.} \\ \mathbf{\text{ centre =}}\mathbf{-}\mathit{a}\mathbf{\text{ \& radius }}\mathbf{=}\sqrt{\mathbf{|}\mathbf{a}{\mathbf{|}}^{\mathbf{2}}\mathbf{-}\mathbf{b}}} \end{gathered}$$