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A197033 Decimal expansion of the shortest distance from the x axis through (2,1) to the line y=x.

%I A197033 #5 Mar 30 2012 18:57:52
%S A197033 1,8,4,4,2,7,1,6,8,1,7,0,0,1,7,1,8,6,4,7,7,9,9,5,7,7,4,4,2,7,3,5,7,0,
%T A197033 2,9,8,4,1,3,4,8,7,6,3,3,8,7,7,0,9,5,0,9,1,5,7,4,7,9,4,0,1,7,8,6,4,8,
%U A197033 7,6,8,3,4,3,8,5,3,8,8,6,1,2,4,8,5,0,6,4,4,7,0,9,9,7,5,8,1,8,5,0
%N A197033 Decimal expansion of the shortest distance from the x axis through (2,1) to the line y=x.
%C A197033 For discussions and guides to related sequences, see A197032, A197008 and A195284.
%e A197033 length of Philo line:  1.8442716817001718647799577442735702984134...
%e A197033 endpoint on x axis:  (2.35321..., 0); see A197032
%e A197033 endpoint on line y=x:  (1.73898, 1.73898)
%t A197033 f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t A197033 g[t_] := D[f[t], t]; Factor[g[t]]
%t A197033 p[t_] := h^2 k + k^3 - h^3 m - h k^2 m - 3 h k t + 3 h^2 m t + 2 k t^2 - 3 h m t^2 + m t^3 (* root of p[t] minimizes f *)
%t A197033 m = 1; h = 2; k = 1; (* m=slope; (h,k)=point *)
%t A197033 t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t A197033 RealDigits[t]  (* A197032 *)
%t A197033 {N[t], 0} (* lower endpoint of minimal segment [Philo line] *)
%t A197033 {N[k*t/(k + m*t - m*h)],
%t A197033 N[m*k*t/(k + m*t - m*h)]} (* upper endpoint *)
%t A197033 d = N[Sqrt[f[t]], 100]
%t A197033 RealDigits[d] (* A197033 *)
%t A197033 Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2.5}],
%t A197033  ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 3}, {y, 0, 3}], PlotRange -> {0, 2}, AspectRatio -> Automatic]
%Y A197033 Cf. A197032, A197008, A195284.
%K A197033 nonn,cons
%O A197033 1,2
%A A197033 _Clark Kimberling_, Oct 10 2011