A circle touches the x-axis and also touches the circle with centre at $(0,3)$ and radius $2$. The locus of the centre of the circle is:

Let the centre of the variable circle be $(h,k)$ and its radius be $r$.Since it touches the x-axis, its radius $r=|k|$.The given circle has centre $C_1(0,3)$ and radius $r_1=2$.The distance between the centres of two touching circles is equal to the sum or difference of their radii.Case 1: External touch.$\sqrt{(h-0{)}^2+(k-3{)}^2}=k+2$$h^2+k^2-6k+9=k^2+4k+4$$h^2=10k-5⟹x^2=10(y-1/2)$ This is the equation of a parabola.