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 November 9th, 2018, 10:38 PM #1 Senior Member   Joined: Jul 2011 Posts: 405 Thanks: 16 Number of rational points The number of rational points on the circle with center $(\sqrt{2},-\sqrt{2})$ and which passes through $(1,-1)$ is
 November 9th, 2018, 11:47 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,968 Thanks: 2217 At a guess, 1. Thanks from topsquark
 November 10th, 2018, 05:10 PM #3 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond For $x=1$ and $y=-1$, $\displaystyle (x-\sqrt2)^2+(y+\sqrt2)^2=6-4\sqrt2$ Generally, if $x$ and $y$ are rational, $\displaystyle x^2-2\sqrt2x+2+y^2+2\sqrt2y+2=6-4\sqrt2$ $\displaystyle x^2+y^2+2\sqrt2(y-x)=2-4\sqrt2$ $\displaystyle \implies x^2+y^2=2$ $\displaystyle \implies x-y=2$ $\displaystyle (y+2)^2+y^2=2$ $\displaystyle 2y^2+4y+2=0\implies y=\frac{-4\pm0}{4}$ $\displaystyle x^2+(x-2)^2=2$ $\displaystyle 2x^2-4x+2=0\implies x=\frac{4\pm0}{4}$ Looks like skipjack's guess is correct. Thanks from topsquark Last edited by greg1313; November 10th, 2018 at 05:44 PM.

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