Equation of given circle

 $$\begin{gathered} {\mathrm{x}}^2+{\mathrm{y}}^2-20\mathrm{x}+4=0 \\ \mathrm{centre} ={\mathrm{c}}_1(10,0), \mathrm{radius}={\mathrm{r}}_1=\sqrt{96} \end{gathered}$$
Let ${\mathrm{c}}_2(\mathrm{x},\mathrm{y})$ be the centre of the circle touching the line x - 2 = 0
$\therefore {\mathrm{r}}_2=|\mathrm{x}-2|$ 
$\therefore$Two circles cut orthogonally
$$\begin{gathered} {\mathrm{c}}_1{\mathrm{c}}_2^2={\mathrm{r}}_1^2+{\mathrm{r}}_2^2 \\ \Rightarrow (\mathrm{x}-10{)}^2+{\mathrm{y}}^2=96+(\mathrm{x}-2{)}^2\Rightarrow {\mathrm{y}}^2=16\mathrm{x} \end{gathered}$$
Locus of centre c₂ is y² =16x