A circle touches the x -axis and also touches the circle with...

Equation of circle with centre $(0,3)$ and radius 2 is $x^2+(y-3)^2=4$ Let locus of the variable circle is $(\alpha, \beta)\because$ It touches $x$-axis. $\therefore$ It equation $(x-\alpha)^2+(y-\beta)^2=\beta^2$ Circles touch externally $$\begin{aligned}& \therefore \sqrt{\alpha^2+(\beta-3)^2}=2+\beta\\ & \alpha^2+(\beta-3)^2=\beta^2+4+4 \beta\\ & \alpha^2=10(\beta-1 / 2) \end{aligned}$$\therefore$Locus is$x^2=10(y-1 / 2) which is parabola.