A circle C touchs the x-axis and the circle $x^2+(y-1{)}^2=1$ externally, then locus of the centre of the circle is given by

$$\begin{gathered} \text{ All the circles having center on the negative y-axis and passing though the } \\ \text{ origin touch the circle }{\mathrm{x}}^2+(\mathrm{y}-1{)}^2=1\text{ and the }\mathrm{x}\text{-axis } \\ \text{ If }(x,y)\text{ is center of the circle }C\text{, then } \end{gathered}$$
$$\begin{gathered} {\mathrm{r}}_1+{\mathrm{r}}_2={\mathrm{A}}_1{ \mathrm{A}}_2\Rightarrow 1+\left|\mathrm{y}\right|=\sqrt{{\mathrm{x}}^2+(\mathrm{y}-1{)}^2} \\ \Rightarrow 1+y^2+2\left|y\right|=x^2+y^2-2y+1 \\ \Rightarrow x^2+2\left|y\right|+2y=0\Rightarrow x^2=4y\text{ if }y\ge 0 \end{gathered}$$
$$\begin{gathered} \text{ Thus locus of the center is } \\ \left\{(\mathrm{x},\mathrm{y})\mid {\mathrm{x}}^2=4\mathrm{y}\right\}\cup \left\{\right(0,\mathrm{y})\mid \mathrm{y}\le 0\} \end{gathered}$$