A circle passes through A(2,1) and touches y-axis then the locus of its centre is 
 

A (2, 1) Touches y-axes 

$$\begin{gathered} {\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0 \\ It touches y-axis then {\mathrm{f}}^2=\mathrm{c} \\ It passes through (2,1) then \\ 4+1+4\mathrm{g}+2\mathrm{f}+\mathrm{c}=0\to \left(1\right) \\ locus of centre (-\mathrm{g}-\mathrm{f})=(\mathrm{x}, \mathrm{y}) \\ from \left(1\right) \\ 5+4(-\mathrm{x})+2(-\mathrm{y})+{\mathrm{y}}^2=0 \\ {\mathrm{y}}^2-2\mathrm{y}=4\mathrm{x}-5 \\ {(\mathrm{y}-1)}^2=4(\mathrm{x}-1) \end{gathered}$$