cp's OEIS Frontend

A197149 Decimal expansion of the shortest distance from the x axis through (1,1) to the line y=3x.

%I A197149 #7 Nov 08 2022 11:41:23
%S A197149 1,9,9,9,1,5,8,3,9,9,5,8,0,3,4,4,2,6,8,8,1,7,4,2,3,5,4,3,8,4,6,1,6,4,
%T A197149 7,3,4,1,3,2,8,2,3,4,7,1,1,8,9,0,6,8,3,7,7,0,9,2,8,1,2,8,1,0,4,6,8,5,
%U A197149 4,8,5,2,8,7,0,0,4,9,0,3,2,7,6,5,3,7,3,7,5,1,3,1,8,9,2,0,5,5,5
%N A197149 Decimal expansion of the shortest distance from the x axis through (1,1) to the line y=3x.
%C A197149 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 A197032, A197008 and A195284.
%e A197149 length of Philo line:    1.999158399580...
%e A197149 endpoint on x axis:    (1.60479, 0); see A197148
%e A197149 endpoint on line y=3x: (0.570212, 1.71064)
%t A197149 f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t A197149 g[t_] := D[f[t], t]; Factor[g[t]]
%t A197149 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 A197149 m = 3; h = 1; k = 1;(* slope m, point (h,k) *)
%t A197149 t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t A197149 RealDigits[t] (* A197148 *)
%t A197149 {N[t], 0} (* endpoint on x axis *)
%t A197149 {N[k*t/(k + m*t - m*h)],
%t A197149 N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=3x *)
%t A197149 d = N[Sqrt[f[t]], 100]
%t A197149 RealDigits[d]  (* A197149 *)
%t A197149 Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2}],
%t A197149 ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 2}, AspectRatio -> Automatic]
%Y A197149 Cf. A197032, A197049, A197008, A195284.
%K A197149 nonn,cons
%O A197149 1,2
%A A197149 _Clark Kimberling_, Oct 11 2011