A circle touches the parabola y=x^2 at the point (1,1) and also passes through the origin. Find the radius of the circle.

Since the circle touches the parabola at $(1,1)$, the radius to that point is along the normal to the parabola there. The slope of the tangent is $2x=2$, so the tangent direction is $(1,2)$ and the normal direction is $(-2,1)$. Hence the centre is $C=(1-2s,\,1+s)$ for some real $s$. The circle passes through the origin, so $|CO|=|C(1,1)|$: $(1-2s)^2+(1+s)^2=(2s)^2+s^2=5s^2$. Expanding, $2-2s+5s^2=5s^2$, so $s=1$ and $C=(-1,2)$. The radius is $|C(1,1)|=\sqrt{4+1}=\sqrt5\approx2.236$. Check: $|CO|=\sqrt{1+4}=\sqrt5$ as well, and $(2,-1)\cdot(1,2)=0$. The radius to two decimals is $2.24$.