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Decimal expansion of the x-intercept of the shortest segment from the x axis through (2,1) to the line y=3x.
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%I #8 Nov 08 2022 12:15:28

%S 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 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 6,0,5,4,3,6,4,2,1,0,3,0,0,7,6,0,4,2

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

%C 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 A root of the polynomial 3*x^3-16*x^2+30*x-25. - _R. J. Mathar_, Nov 08 2022

%e length of Philo line: 3.160946973065...; see A197151

%e endpoint on x axis: (2.85106, 0)

%e endpoint on line y=3x: (0.802397, 2.40719)

%t f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;

%t g[t_] := D[f[t], t]; Factor[g[t]]

%t 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 m = 3; h = 2; k = 1;(* slope m, point (h,k) *)

%t t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]

%t RealDigits[t] (* A197150 *)

%t {N[t], 0} (* endpoint on x axis *)

%t {N[k*t/(k + m*t - m*h)],

%t N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=3x *)

%t d = N[Sqrt[f[t]], 100]

%t RealDigits[d] (* A197151 *)

%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 3}],

%t ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4}, {y, 0, 3}],

%t PlotRange -> {0, 2.5}, AspectRatio -> Automatic]

%Y Cf. A197032, A197151, A197008, A195284.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Oct 11 2011

%E Incorrect trailing digits removed. - _R. J. Mathar_, Nov 08 2022