%I #19 May 03 2024 15:49:44

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

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

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

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

%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 Any Philo line can be constructed using the intersections of circles and hyperbolas. In this case, the Philo line passes though the two points at which the circle (x - 3/2)^2 + (y - 1)^2 = 13/4 and the hyperbola x*y - y^2 = 2 intersect. The circle has segment OP as diameter, where O(0,0) is the origin and P is the point (3,2). The asymptotes of the hyperbola are the x axis and the line y = x. Point P is one of the two points at which the circle and the hyperbola intersect. - _Raul Prisacariu_, Apr 06 2024

%e Length of Philo line: 2.8861171058980012915367...

%e Endpoint on x axis: (3.4883, 0); see A197138

%e Endpoint on line y = x: (2.80376, 2.80376)

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

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

%t RealDigits[t] (* A197138 *)

%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)]} (* endpoint on line y=x *)

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

%t RealDigits[d] (* this sequence *)

%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],

%t ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 4}, {y, 0, 3}],

%t PlotRange -> {0, 3}, AspectRatio -> Automatic]

%Y Cf. A197032, A197138, A197008, A195284.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Oct 10 2011

%E Last digit removed (repr. truncated, not rounded up) by _R. J. Mathar_, Nov 08 2022