Function Repository Resource:

FociPointHyperbola

Find the equation for a hyperbola given two foci and a point

Contributed by: Ed Pegg Jr and Jan Mangaldan

ResourceFunction["FociPointHyperbola"][{f₁,f₂,p},{x,y}]

returns the hyperbola A x²+B x y+C y²+D x+E y+F in the variables x and y, given the foci f₁,f₂ and a point p through which the hyperbola passes.

ResourceFunction["FociPointHyperbola"][{f₁,f₂,p},t]

returns a parametric equation in the variable t.

Examples

Basic Examples (4)

Find the Cartesian equation of a hyperbola with foci (2,2) and (3,5) that goes through point (1,3):

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Show the hyperbola:

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Find the three hyperbolas generated by the three points:

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The three hyperbolas happen to intersect at the inner and outer Soddy centers for triangle ΔFGH:

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Scope (2)

Generate the parametric equations of a hyperbola with foci (2,2) and (3,5) that goes through point (1,3):

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Show the hyperbola:

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$ParametricPlot[hyp, {t, -\[Pi], \[Pi]}, Axes -> None, Epilog -> {Inset["F", F], Inset["G", G], Inset["H", H]}, Frame -> True]$

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Properties and Relations (4)

Use a different set of variables:

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Use formal variables:

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$ResourceFunction[ "FociPointHyperbola"][{{0, 0}, {5, 7}, {2, 1}}, {\[FormalX], \[FormalY] }]$

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If no variables are given, formal variables are used by default:

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Use FociPointHyperbola to generate the implicit Cartesian equation of a hyperbola:

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{F, G, H} = {{8, 0}, {0, 6}, {18/5, 33/10}};
ResourceFunction["FociPointHyperbola"][{F, G, H}, {x, y}]

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Use GroebnerBasis to get an equivalent result:

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Generate an equivalent parametric equation:

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Use GroebnerBasis to derive the implicit Cartesian equation from the parametric equation:

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$First[GroebnerBasis[ Append[Thread[{x, y} == %], Sec[t]^2 - Tan[t]^2 == 1], {x, y}, {Sec[t], Tan[t]}]]$

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Use the resource function HyperbolaProperties to generate properties of the hyperbola:

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Get a hyperbola equation:

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Show positions for coefficients in A x²+B x y+C y²+D x+E y+F=0:

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$pos = CoefficientList[\[FormalCapitalA] x^2 + \[FormalCapitalB] x y + \[FormalCapitalC] y^2 + \[FormalCapitalD] x + \[FormalCapitalE] y + \[FormalCapitalF], {x, y}]$

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Get the coefficients:

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See the coefficients in the standard order:

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Neat Examples (5)

Three arbitrary circles and some middle circles:

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circs = {Circle[{0, 0}, 1], Circle[{8, 0}, 3], Circle[{0, 6}, 2]};
Graphics[{circs, Green, Circle[{3, 0}, 2], Circle[{0, 5/2}, 3/2], Circle[{18/5, 33/10}, 5/2]}]

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Find the three hyperbolas using two circle centers and a midcircle center:

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ab = ResourceFunction[
"FociPointHyperbola"][{{0, 0}, {8, 0}, {3, 0}}, {x, y}];
ac = ResourceFunction[
"FociPointHyperbola"][{{0, 0}, {0, 6}, {0, 5/2}}, {x, y}];
bc = ResourceFunction[
"FociPointHyperbola"][{{8, 0}, {0, 6}, {18/5, 33/10}}, {x, y}];

Find real-valued intersection points of the three hyperbolas:

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Find a few distances:

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Show two circles tangent to the given circles:

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Version History

1.0.0 – 18 July 2022

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License Information

This work is licensed under a Creative Commons Attribution 4.0 International License