cp's OEIS Frontend

A197025 Decimal expansion of the radius of the circle tangent to the curve y=3/(1+x^2) and to the positive x and y axes.

%I A197025 #5 Mar 30 2012 18:57:52
%S A197025 7,0,9,9,1,4,9,2,6,8,6,0,4,3,9,9,2,1,4,0,9,5,9,3,8,8,9,1,6,2,5,5,8,8,
%T A197025 7,3,4,8,0,0,2,9,4,2,6,7,8,4,6,5,6,6,5,4,0,4,6,9,1,2,2,1,1,0,5,0,6,6,
%U A197025 9,0,1,8,1,5,4,2,7,7,6,0,0,7,2,4,9,1,9,4,6,5,3,3,2,0,2,1,1,5
%N A197025 Decimal expansion of the radius of the circle tangent to the curve y=3/(1+x^2) and to the positive x and y axes.
%C A197025 Let (x,y) denote the point of tangency.  Then
%C A197025 x=1.24529475848935327220740793075336892358...
%C A197025 y=1.17612050304358700012836336674764176692...
%C A197025 slope=-1.148377151582141158728147357192644...
%C A197025 (The Mathematica program includes a graph.)
%e A197025 radius=0.70991492686043992140959388916255887...
%t A197025 r = .71; c = 3;
%t A197025 Show[Plot[c/(1 + x^2), {x, 0, 2}],
%t A197025  ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, 0, 2}, {y, 0, 2}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]
%t A197025 u[x_] := (x*(1 + x^2)^3 - 2*x*c^2)/((1 + x^2)^3 - 2*c*x*(1 + x^2))
%t A197025 v = x /. FindRoot[c/(1 + x^2) == u[x] + Sqrt[2*u[x]*x - x^2], {x, .9, 1.45}, WorkingPrecision -> 100]
%t A197025 t = Re[v]; RealDigits[t] (* x coord. of tangency pt. *)
%t A197025 y = c/(1 + t^2)          (* y coord. of tangency pt. *)
%t A197025 radius = u[t]
%t A197025 RealDigits[radius] (* A197025 *)
%t A197025 slope = -2*c*t/(1 + t^2)^2   (* slope at tangency pt *)
%Y A197025 Cf. A197023, A196024.
%K A197025 nonn,cons
%O A197025 1,1
%A A197025 _Clark Kimberling_, Oct 08 2011