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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### Properties and Relations (4)

Use a different set of variables:

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Use formal variables:

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

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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