login
Decimal expansion of the radius of the circle tangent to the curve y=1/(1+x^2) and to the positive x and y axes.
3

%I #9 Mar 30 2012 18:57:52

%S 3,9,5,8,6,2,4,3,7,8,4,7,4,8,0,7,9,8,2,4,0,7,0,7,7,2,2,5,6,6,3,1,5,5,

%T 2,7,3,3,4,3,4,3,9,2,9,2,4,9,1,0,0,0,8,2,5,0,4,4,2,7,9,6,3,2,5,1,2,6,

%U 4,4,3,7,9,1,1,1,8,8,1,3,0,8,6,6,5,3,8,4,8,9,0,0,5,9,1,5,8,3,6

%N Decimal expansion of the radius of the circle tangent to the curve y=1/(1+x^2) and to the positive x and y axes.

%C Let (x,y) denote the point of tangency. Then

%C x=0.611116305530271280094961817959748153764566...

%C y=0.728086522083031555694579423798015759485165...

%C slope=-0.64791770945231344102369199673001177755...

%C (The Mathematica program includes a graph.)

%e radius=0.39586243784748079824070772256631552733434...

%t r = .396; c = 1;

%t Show[Plot[c/(1 + x^2), {x, 0, 1.5}],

%t ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}],

%t PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]

%t u[x_] := (x*(1 + x^2)^3 - 2*x*c^2)/((1 + x^2)^3 - 2*c*x*(1 + x^2))

%t v = x /. FindRoot[c/(1 + x^2) == u[x] + Sqrt[2*u[x]*x - x^2], {x, .4, 1},

%t WorkingPrecision -> 100]

%t t = Re[v] ; RealDigits[t] (* x coord. of tangency pt. *)

%t y = c/(1 + t^2) (* y coord. of tangency pt. *)

%t radius = u[t]

%t RealDigits[radius] (* A197023 *)

%t slope = -2*c*t/(1 + t^2)^2 (* slope at tangency point *)

%Y Cf. A197020, A197024, A197025, A197016.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Oct 08 2011