A197024 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, 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, 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, 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

Offset: 0

Author

Clark Kimberling, Oct 08 2011

Keywords

• nonn
• cons

Comments

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

x=0.290074091667981539080192147132694221247...

y=0.461193781487549868098884143492334039544...

slope=-0.24679469383945033223474847695422791...

(The Mathematica program includes a graph.)

Examples

			radius=0.23400514059513790173472762376722996062...
		

Crossrefs

Cf. A197023, A197025.

Programs

Mathematica

r = .234; c = 1/2;
Show[Plot[c/(1 + x^2), {x, 0, 0.8}],
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] (* A197024 *)
slope = -2*c*t/(1 + t^2)^2  (* slope at tangency pt. *)