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A197143
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Decimal expansion of the shortest distance from the x axis through (2,1) to the line y=2x.
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3
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2, 7, 4, 6, 3, 9, 4, 1, 0, 7, 6, 1, 0, 0, 7, 1, 1, 6, 5, 6, 7, 9, 9, 5, 4, 9, 7, 2, 2, 5, 2, 5, 7, 3, 3, 7, 4, 3, 9, 0, 5, 1, 4, 5, 6, 9, 1, 4, 5, 8, 6, 7, 1, 7, 4, 6, 4, 6, 3, 3, 5, 2, 3, 4, 4, 2, 2, 7, 3, 4, 8, 3, 1, 6, 8, 3, 0, 0, 4, 7, 0, 6, 1, 1, 5, 0, 0, 9, 6, 4, 4, 3, 2, 2, 4, 7, 9, 5, 1
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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length of Philo line: 2.7463941076100...
endpoint on x axis: (2.69141, 0); see A197142
endpoint on line y=2x: (1.1295, 2.25901)
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MATHEMATICA
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f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
g[t_] := D[f[t], t]; Factor[g[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
m = 2; h = 2; k = 1; (* slope m, point (h, k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
{N[t], 0} (* endpoint on x axis *)
{N[k*t/(k + m*t - m*h)],
N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=2x *)
d = N[Sqrt[f[t]], 100]
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],
ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 2.5}, AspectRatio -> Automatic]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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