A variable circle passes through the fixed point (2, 0) and touches y-axis then the locus of its centre is 
 

$$\begin{gathered} \mathrm{S}\equiv {\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0 \\ it \mathrm{passes} \mathrm{through} (2, 0) then \\ 4+4\mathrm{g}+\mathrm{c}=0\to \left(1\right) \\ \mathrm{and} \mathrm{touches} \mathrm{y}-\mathrm{axes} \\ {\mathrm{f}}^2=\mathrm{c} \\ locus of centre(-\mathrm{g}, -\mathrm{f})=(\mathrm{x}, \mathrm{y}) \\ -\mathrm{g}=\mathrm{x}, -\mathrm{f}=\mathrm{y} \\ from \left(1\right),4-4\mathrm{x}+{\mathrm{y}}^2=0 \\ \Rightarrow {\mathrm{y}}^2=4\mathrm{x}-4 \\ It represents \mathrm{parabola} \end{gathered}$$