cp's OEIS Frontend

A197150 Decimal expansion of the x-intercept of the shortest segment from the x axis through (2,1) to the line y=3x.

%I A197150 #8 Nov 08 2022 12:15:28
%S A197150 2,8,5,1,0,5,9,7,9,0,6,2,9,8,6,2,6,3,9,8,2,6,0,7,2,3,4,0,8,5,3,6,2,6,
%T A197150 8,9,3,3,3,9,0,9,3,5,8,1,7,2,2,1,0,1,7,3,2,2,7,0,3,3,5,9,2,6,3,9,6,4,
%U A197150 6,0,5,4,3,6,4,2,1,0,3,0,0,7,6,0,4,2
%N A197150 Decimal expansion of the x-intercept of the shortest segment from the x axis through (2,1) to the line y=3x.
%C A197150 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.
%C A197150 A root of the polynomial 3*x^3-16*x^2+30*x-25. - _R. J. Mathar_, Nov 08 2022
%e A197150 length of Philo line:    3.160946973065...; see A197151
%e A197150 endpoint on x axis:    (2.85106, 0)
%e A197150 endpoint on line y=3x: (0.802397, 2.40719)
%t A197150 f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t A197150 g[t_] := D[f[t], t]; Factor[g[t]]
%t A197150 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 A197150 m = 3; h = 2; k = 1;(* slope m, point (h,k) *)
%t A197150 t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t A197150 RealDigits[t]  (* A197150 *)
%t A197150 {N[t], 0} (* endpoint on x axis *)
%t A197150 {N[k*t/(k + m*t - m*h)],
%t A197150 N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=3x *)
%t A197150 d = N[Sqrt[f[t]], 100]
%t A197150 RealDigits[d]  (* A197151 *)
%t A197150 Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 3}],
%t A197150 ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4}, {y, 0, 3}],
%t A197150 PlotRange -> {0, 2.5}, AspectRatio -> Automatic]
%Y A197150 Cf. A197032, A197151, A197008, A195284.
%K A197150 nonn,cons
%O A197150 1,1
%A A197150 _Clark Kimberling_, Oct 11 2011
%E A197150 Incorrect trailing digits removed. - _R. J. Mathar_, Nov 08 2022