3. Postdoctoral Program - MSRI

Figure 3: The hyperbola 2xy + x + y + 2 = 0
2.7 Rational Distance Sets on the Hyperbola
Theorem 6. Let a, b, c, d ∈ Q such that ad − bc �= 0, then there exists a rational
distance set of three rational points, S = {P1, P2, P3} on the hyperbola axy + bx +
cy + d = 0.
Proof. The proof will proceed in several parts. First, we rewrite the distance formula
as in equation 1. So, yi−yj
xi−xj
can be paramterized like yi−yj
xi−xj = m2 ij −1
2mij .
Now, we can solve for yi in our equation for the hyperbola to get the expression:
yi = − bxi+d
axi+c . We then substitute yi into our modified distance formula and this gives
the equation
ad−bc
(axi+c)(axj+c)
= yi−yj
xi−xj = m2 ij −1
2mij .
Let D = ad − bc. If we take m2 −1
2m = Dn2 and make the substitutions m = X
2D and
n = Y
2DX we get the elliptic curve E (D) : Y 2 = X 3 − 4D 2 X. We can also obtain the
expression m2 −1
2m = Dn2 from E (D) with the substitutions X = 2Dm and Y = 4D 2 mn.
Now, if we let Qi = (Xi: Yi: 1) be rational points on E (D) , then we define the
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