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 
 

$$\begin{gathered} Let C_1(h,k) be the center of the circle. \\ Circle touches the x-axis then its radius is r_1=k. \\ Also circle touches the circle with centre C_2(0,3) and radius r_2=2. \\ \therefore C_1{C}_2=r_1+r_2 \Rightarrow {(h-0)}^2+{(k-3)}^2 ={(k+2)}^2 Squaring on both sides \\ {h}^2-10k+5=0 \\ \Rightarrow Locus is x^2-10y+5=0, which is parabola. \end{gathered}$$