Circles with centre in the first quadrant touch both coordinate axes. Consider those that also touch the circle x^2+y^2=1 (the contact may be…

A circle touching both axes in the first quadrant has centre $(r,r)$ and radius $r$, so its centre is at distance $\sqrt2\,r$ from the origin. It touches the unit circle (centre $O$, radius $1$) when the distance between centres equals $1+r$ (external contact) or $|1-r|$ (internal contact). External: $\sqrt2r=1+r$, so $r=\dfrac1{\sqrt2-1}=\sqrt2+1$. Internal with the small circle inside the unit circle: $\sqrt2r=1-r$, so $r=\dfrac1{\sqrt2+1}=\sqrt2-1\approx0.414$. Internal with the unit circle inside the other circle: $\sqrt2r=r-1$ gives $r<0$, impossible. The valid radii are $\sqrt2+1$ and $\sqrt2-1$, with sum $2\sqrt2\approx2.83$. Substituting both values back confirms each distance condition.