A197147 - OEIS

%I #6 Mar 30 2012 18:57:52

%S 4,7,0,8,0,0,0,0,1,7,4,9,6,4,6,0,2,7,3,9,3,1,7,3,5,6,4,4,1,0,5,4,5,3,

%T 5,3,3,8,5,0,6,9,2,6,7,9,9,5,1,2,9,0,8,3,1,2,1,0,9,5,6,9,5,1,9,1,4,2,

%U 6,9,5,3,3,3,0,7,7,9,3,1,2,8,6,3,1,3,7,8,1,7,5,8,5,6,3,2,3,5,5

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

%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.

%e length of Philo line:    4.708000017496..

%e endpoint on x axis:    (4.92546, 0); see A197146

%e endpoint on line y=2x: (1.72768, 3.45536)

%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 = 2; h = 4; k = 1;  (* slope m, point (h,k) *)

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

%t RealDigits[t]   (* A197146 *)

%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=2x *)

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

%t RealDigits[d]   (* A197147 *)

%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 5}],

%t ContourPlot[(x - h)^2 + (y - k)^2 == .004, {x, 0, 5}, {y, 0, 3}], PlotRange -> {0, 4}, AspectRatio -> Automatic]

%Y Cf. A197032, A197146, A197008, A195284.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Oct 11 2011