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Saturday, April 03, 2010

Ellipses, Hyperbolas and Pell's Equation

Some simple algebra can connect ellipses and hyperbolas, two conic sections, with the equation x2 - Dy2 = 1, which is commonly known as Pell's Equation.

While Pell's Equation is usually considered as a Diophantine equation, it is easier to let it be rational, so what follows is in the field of rationals, and my first relational equation is:

(D-1)j2 + (j+/-1)2 = (x+y)2

where j = ((x+Dy) -/+1)/D

Here the +/- indicates that one variation will work so it is an OR and not an AND. Either plus OR minus will give a valid j.

(For the derivation go to an earlier post.)

Notice you get Pythagorean Triples if D-1 is a perfect square! So immediately Pell's Equation is connected to circles.

Here's an example.

With D=2, and x=17, y=12, you solve Pell's as 172 - 2(12)2 = 1, and going with the minus of the plus or minus:

j = ((17+2(12)-1)/2 = 20 is a solution giving:

202 + 212 = 292

So D-1 a perfect square relates to circles but otherwise gives non-circular ellipses!

So you can always find integer solutions for af2 + g2 = h2 by using a solution for Pell's Equation where D = a+1.

For example with D=3, x=2, y=1 works as 22 - 3(1)2 = 1.

And going with the plus of the plus or minus:

j = ((2+3(1)) +1 )/3 = 2 is a solution giving, 2(2)2 + 12 = 32

But I've set the field as rationals which allows me to bring in one other result known since antiquity which is the rational parametric solution for Pell's Equation:

y = 2t/(D - t2) and x = (D + t2)/(D - t2)

(For my re-derivation go to an earlier post.)

So you can always find integer solutions for the ellipse using the single parameter t.

But in rationals x2 - Dy2 = 1, where D is a positive integer will give you hyperbolas, but for each solution for x and y, you get an ellipse, so there is a direct connection between hyperbolas as ellipse generators.

So one can directly connect ellipses to hyperbolas using Pell's Equation as a hyperbola itself in rationals and get Diophantine solutions for the ellipses as well, so the rational hyperbola can generate integer solutions to ellipses, including for the special case D-1, Pythagorean Triples.

Beautifully simple relations between two of the conic sections.


James Harris
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