OFFSET
0,1
COMMENTS
Let (x,y) denote the point of tangency. Then
x=0.611116305530271280094961817959748153764566...
y=0.728086522083031555694579423798015759485165...
slope=-0.64791770945231344102369199673001177755...
(The Mathematica program includes a graph.)
EXAMPLE
radius=0.39586243784748079824070772256631552733434...
MATHEMATICA
r = .396; c = 1;
Show[Plot[c/(1 + x^2), {x, 0, 1.5}],
ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}],
PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]
u[x_] := (x*(1 + x^2)^3 - 2*x*c^2)/((1 + x^2)^3 - 2*c*x*(1 + x^2))
v = x /. FindRoot[c/(1 + x^2) == u[x] + Sqrt[2*u[x]*x - x^2], {x, .4, 1},
WorkingPrecision -> 100]
t = Re[v] ; RealDigits[t] (* x coord. of tangency pt. *)
y = c/(1 + t^2) (* y coord. of tangency pt. *)
radius = u[t]
RealDigits[radius] (* A197023 *)
slope = -2*c*t/(1 + t^2)^2 (* slope at tangency point *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Oct 08 2011
STATUS
approved