Let $z\in C$, the set of complex numbers. Then the equation, $2\left|\mathrm{z} + 3\mathrm{i}\right| - \left|\mathrm{z}-\mathrm{i}\right| = 0$ represents 

$$\begin{gathered} 2|\mathrm{z}+3\mathrm{i}|=|\mathrm{z}-\mathrm{i}| \\ \mathrm{Let} \mathrm{z}=\mathrm{x}+\mathrm{iy} \\ \Rightarrow 2|\mathrm{x}+\mathrm{iy}+3\mathrm{i}|=|\mathrm{x}+\mathrm{iy}-\mathrm{i}| \\ \Rightarrow 2|\mathrm{x}+\mathrm{i}(\mathrm{y}+3\left)\right|=|\mathrm{x}+\mathrm{i}(\mathrm{y}-1\left)\right| \\ \Rightarrow 4\left({\mathrm{x}}^2+(\mathrm{y}+3{)}^2\right)={\mathrm{x}}^2+(\mathrm{y}-1{)}^2 \\ \Rightarrow 4{\mathrm{x}}^2+4\left({\mathrm{y}}^2+6\mathrm{y}+9\right)={\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{y}+1 \\ \Rightarrow 3{\mathrm{x}}^2+3{\mathrm{y}}^2+26\mathrm{y}+35=0 \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2+\frac{26}{3}\mathrm{y}+\frac{35}{3}=0 \end{gathered}$$

represents a circle with radius=

$$\begin{gathered} \mathrm{r}=\sqrt{\frac{169}{9}-\frac{35}{5}}=\sqrt{\frac{169-105}{9}}=\sqrt{\frac{64}{9}}=\frac{8}{3}. \\ \boxed{\mathrm{r}=\frac{8}{3}} \end{gathered}$$