cp's OEIS Frontend

A197141 Decimal expansion of the shortest distance from the x axis through (1,1) to the line y=2x.

%I A197141 #8 Nov 08 2022 11:37:35
%S A197141 1,6,7,3,6,4,7,3,0,4,1,5,2,9,1,5,0,7,8,0,1,3,8,6,3,4,3,3,2,7,8,1,6,6,
%T A197141 0,2,6,8,5,8,3,6,5,7,7,1,0,3,5,3,9,2,8,6,1,7,9,9,4,6,0,5,6,9,5,2,6,1,
%U A197141 8,9,5,6,2,8,0,5,4,7,5,7,2,9,1,1,9,3,7,1,7,0,9,5,8,5,1,2,9,5,3
%N A197141 Decimal expansion of the shortest distance from the x axis through (1,1) to the line y=2x.
%C A197141 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 A197141 length of Philo line:    1.6736473041529...
%e A197141 endpoint on x axis:    (1.44062, 0); see A197140
%e A197141 endpoint on line y=2x: (0.765782, 1.53156)
%t A197141 f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t A197141 g[t_] := D[f[t], t]; Factor[g[t]]
%t A197141 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 m = 2; h = 1; k = 1; (* slope m, point (h,k) *)
%t A197141 t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t A197141 RealDigits[t]  (* A197140 *)
%t A197141 {N[t], 0} (* endpoint on x axis *)
%t A197141 {N[k*t/(k + m*t - m*h)],
%t A197141 N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=mx *)
%t A197141 d = N[Sqrt[f[t]], 100]
%t A197141 RealDigits[d] (* A197141 *)
%t A197141 Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2}],
%t A197141 ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 1.7}, AspectRatio -> Automatic]
%Y A197141 Cf. A197032, A197140, A197008, A195284.
%K A197141 nonn,cons
%O A197141 1,2
%A A197141 _Clark Kimberling_, Oct 11 2011