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.
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, 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, 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
Offset: 1
Examples
radius=0.70991492686043992140959388916255887...
Programs
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Mathematica
r = .71; c = 3; Show[Plot[c/(1 + x^2), {x, 0, 2}], ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, 0, 2}, {y, 0, 2}], 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, .9, 1.45}, 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] (* A197025 *) slope = -2*c*t/(1 + t^2)^2 (* slope at tangency pt *)
Comments