The equation $\left|\sqrt{\mathrm{x}-2{)}^2+{\left(\mathrm{y}-1\right)}^2}-\sqrt{{\left(\mathrm{x}+2\right)}^2+{\mathrm{y}}^2}\right|=\mathrm{c}$ will represents a hyperbola if 
 

Given $$\begin{gathered} \left|\sqrt{{\left(\mathrm{x}-2\right)}^2+{\left(\mathrm{y}-1\right)}^2}-\sqrt{{(x+2)}^2+y^2}\right|=c \\ It is in the form of \left|SP-S\text{'}P\right|=2a \end{gathered}$$

from this equation  we get the coordinates of two focii are $S=(2, 1), S\text{'}=(-2, 0) and P=(\mathrm{x}, \mathrm{y})$,c=2a

Also $SS\text{'} > 2\mathrm{a}$

$$\begin{gathered} \Rightarrow \sqrt{4^2+1^2} > 2\mathrm{a} \\ \Rightarrow 2\mathrm{a} < \sqrt{17} \\ \Rightarrow c< \sqrt{17} (\because c=2a) \\ \therefore c\in \left(0, \sqrt{17}\right) \end{gathered}$$