cp's OEIS Frontend

A197155 Decimal expansion of the shortest distance from the x axis through (4,1) to the line y=x/2.

%I A197155 #9 Feb 16 2025 02:48:46
%S A197155 1,9,4,6,3,4,6,4,0,2,3,7,8,4,8,3,8,5,6,1,6,6,4,0,9,1,1,4,2,3,0,0,8,0,
%T A197155 6,8,1,8,5,8,2,1,0,6,7,1,1,7,6,0,3,8,5,7,0,1,8,9,2,3,8,5,0,9,1,0,4,9,
%U A197155 9,8,9,5,6,0,1,8,8,6,8,0,1,9,1,0,7,7,4,4,3,2,0,7,0,6,5,2,2,4,1,4
%N A197155 Decimal expansion of the shortest distance from the x axis through (4,1) to the line y=x/2.
%C A197155 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 A197155 length of Philo line:  1.94634640...
%e A197155 endpoint on x axis:    (4.2236, 0); see A197154
%e A197155 endpoint on line y=3x: (3.79888, 1.89944)
%t A197155 f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t A197155 g[t_] := D[f[t], t]; Factor[g[t]]
%t A197155 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 A197155 m = 1/2; h = 4; k = 1;(* slop m, point (h,k) *)
%t A197155 t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t A197155 RealDigits[t]  (* A197154 *)
%t A197155 {N[t], 0} (* endpoint on x axis *)
%t A197155 {N[k*t/(k + m*t - m*h)],
%t A197155  N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=x/2 *)
%t A197155 d = N[Sqrt[f[t]], 100]
%t A197155 RealDigits[d]  (* A197155 *)
%t A197155 Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4.5}],
%t A197155  ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4.5}, {y, 0, 3}],
%t A197155  PlotRange -> {0, 2}, AspectRatio -> Automatic]
%Y A197155 Cf. A197032, A197155, A197008, A195284.
%K A197155 nonn,cons
%O A197155 1,2
%A A197155 _Clark Kimberling_, Oct 11 2011