NIMCET 2023 Question Paper with solutions

Let ${\mathrm{C}}_1\left(\mathit{h},\mathit{k}\right)$ be the center of the circle. Circle touches the $\mathit{x}$-axis then its radius is ${\mathit{r}}_1=\mathit{k}$
Also circle touches the circle with centre ${\mathrm{C}}_2$ $\left(0,3\right)$ and radius ${\mathit{r}}_2=2$
$\therefore \left|{\mathrm{C}}_1{\mathrm{C}}_2\right|={\mathit{r}}_1+{\mathit{r}}_2$
$\Rightarrow \sqrt{{\left(\mathit{h}-0\right)}^2+{\left(\mathit{k}-3\right)}^2}=|\mathit{k}+2|$
$\Rightarrow {\mathit{h}}^2-10\mathit{k}+5=0$
Change $\mathit{h}$ to $\mathit{x}$ and $\mathit{k}$ to $\mathit{y}$
$\Rightarrow {\mathit{x}}^2-10\mathit{y}+5=0$
It is a parabola.