cp's OEIS Frontend

A197035 Decimal expansion of the shortest distance from the x axis through (3,1) to the line y=x.

%I A197035 #9 Mar 30 2012 18:57:52
%S A197035 2,6,0,8,1,9,4,0,2,4,9,6,1,0,1,8,9,0,1,9,9,0,1,4,4,5,4,2,8,3,5,2,2,3,
%T A197035 9,5,9,0,8,3,5,8,9,1,1,5,8,7,9,5,9,7,6,7,4,4,9,4,9,1,7,7,5,6,3,8,5,7,
%U A197035 7,4,4,9,2,8,8,4,1,8,9,9,6,8,0,3,9,1,0,4,3,1,5,5,9,0,5,1,4,5,0
%N A197035 Decimal expansion of the shortest distance from the x axis through (3,1) to the line y=x.
%C A197035 The shortest segment from one side of an angle T through a point P inside T is called the Philo line of P in T.  For discussions and guides to related sequences, see A197008 and A195284.
%e A197035 length of Philo line:  2.60819402496101...
%e A197035 endpoint on x axis:   (3.47797, 0); see A197034
%e A197035 endpoint on line y=x: (2.35321, 2.35321)
%t A197035 f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t A197035 g[t_] := D[f[t], t]; Factor[g[t]]
%t A197035 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
%t A197035 m = 1; h = 3; k = 1;  (* slope m; point (h,k) *)
%t A197035 t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t A197035 RealDigits[t]  (* A197034 *)
%t A197035 {N[t], 0} (* endpoint on x axis *)
%t A197035 {N[k*t/(k + m*t - m*h)], N[m*k*t/(k + m*t - m*h)]} (* upper endpoint *)
%t A197035 d = N[Sqrt[f[t]], 100]
%t A197035 RealDigits[d] (* A197035 *)
%t A197035 Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],
%t A197035 ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 3.5}, {y, 0, 3}], PlotRange -> {0, 3}, AspectRatio -> Automatic]
%Y A197035 Cf. A197032, A197034, A197008, A195284.
%K A197035 nonn,cons
%O A197035 1,1
%A A197035 _Clark Kimberling_, Oct 10 2011