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A197139
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Decimal expansion of the shortest distance from the x axis through (3,2) to the line y = x.
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3
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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, 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, 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
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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.
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
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LINKS
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EXAMPLE
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Length of Philo line: 2.8861171058980012915367...
Endpoint on x axis: (3.4883, 0); see A197138
Endpoint on line y = x: (2.80376, 2.80376)
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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 = 1; h = 3; k = 2; (* 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)]} (* endpoint on line y=x *)
d = N[Sqrt[f[t]], 100]
RealDigits[d] (* this sequence *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],
ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 4}, {y, 0, 3}],
PlotRange -> {0, 3}, AspectRatio -> Automatic]
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Last digit removed (repr. truncated, not rounded up) by R. J. Mathar, Nov 08 2022
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STATUS
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approved
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