A parabola touches $x-$axis at $\left(1,0\right)$ and $y-$axis at $\left(0,2\right)$. Then which is/are correct?

Focus : $\left(\frac{{\mathrm{ab}}^2}{{\mathrm{a}}^2+{\mathrm{b}}^2},\frac{{\mathrm{a}}^2\mathrm{b}}{{\mathrm{a}}^2+{\mathrm{b}}^2}\right)$ if parabola meets
x – axis at (a, 0) and y – axis at (0, b)
a = 1, b = 2 $\Rightarrow \mathrm{S}=\left(\frac{4}{5},\frac{2}{5}\right)$
Vertex $\left(\frac{{\displaystyle {\mathrm{ab}}^4}}{{\displaystyle {({\mathrm{a}}^2+{\mathrm{b}}^2)}^2}},\frac{{\displaystyle {\mathrm{a}}^4\mathrm{b}}}{{\displaystyle {({\mathrm{a}}^2+{\mathrm{b}}^2)}^2}}\right)=\left(\frac{{\displaystyle 16}}{{\displaystyle 25}},\frac{{\displaystyle 2}}{{\displaystyle 25}}\right)$