Osculating circles in ellipse

Hi,
I am using Wolfram Mathematica for many things, but today I spent lots of time finding the radius of a circle in osculation circles if the ellipse has o1= 4 o2=8. I tried lots of combinations in Mathematica, but don’t know how to make it.

(radius from K to A)

Computation attached. See http://mathworld.wolfram.com/RadiusofCurvature.html