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

%I #8 Mar 30 2012 18:57:52

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

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

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

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

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

%C x=0.290074091667981539080192147132694221247...

%C y=0.461193781487549868098884143492334039544...

%C slope=-0.24679469383945033223474847695422791...

%C (The Mathematica program includes a graph.)

%e radius=0.23400514059513790173472762376722996062...

%t r = .234; c = 1/2;

%t Show[Plot[c/(1 + x^2), {x, 0, 0.8}],

%t ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], 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}, 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] (* A197024 *)

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

%Y Cf. A197023, A197025.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Oct 08 2011