Explanation:
C Given that, The circle with centre \(\mathrm{C}_1\) touches the another circle with centre \(\mathrm{C}_2(0,3)\) and radius \(\mathrm{r}_2=2\) \(\therefore\left|\mathrm{C}_1 \mathrm{C}_2\right|=\mathrm{r}_1+\mathrm{r}_2\) Let \(\mathrm{C}_1(\mathrm{~h}, \mathrm{k})\) be the centre of the circle. \(\because \mathrm{Cir}_2\). touches the \(\mathrm{x}\)-axis then its radius is \(\mathrm{r}_1=\mathrm{k}\). \(\sqrt{(\mathrm{h}-0)^2+(\mathrm{k}-3)^2}=(\mathrm{k}+2)\) on squaring both side, we get :- \(\mathrm{h}^2-10 \mathrm{k}+5=0\) Now replace \(\mathrm{h}\) with \(\mathrm{x}\) and \(\mathrm{k}\) with \(\mathrm{y}\) in the above equation we get Locus is \(\mathrm{x}^2-10 \mathrm{y}+5=0\), which is parabola.
