Let $r$ be the required radius
As origin lies on the circle, the distance of $\left(0,0\right)$ from $\left(0,y\right)=r$ 

$\therefore \text{y}=\text{r}$



As the two circles touch externally, distance between centres is sum of the radii

$\Rightarrow \sqrt{{\left(0-1\right)}^2+{\left(\mathrm{y}-1\right)}^2}=r+1$

$\Rightarrow 1+y^2-2y+1=r^2+2r+1$

$\Rightarrow 1+r^2-2r=r^2+2r$, $\left[\because y=r\right]$

$\therefore 1=4r$

$\therefore r=\frac{1}{4}$