November 9th, 2018, 11:38 PM  #1 
Senior Member Joined: Jul 2011 Posts: 402 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 10th, 2018, 12:47 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,310 Thanks: 1980 
At a guess, 1.

November 10th, 2018, 06:10 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,913 Thanks: 1113 Math Focus: Elementary mathematics and beyond 
For $x=1$ and $y=1$, $\displaystyle (x\sqrt2)^2+(y+\sqrt2)^2=64\sqrt2$ Generally, if $x$ and $y$ are rational, $\displaystyle x^22\sqrt2x+2+y^2+2\sqrt2y+2=64\sqrt2$ $\displaystyle x^2+y^2+2\sqrt2(yx)=24\sqrt2$ $\displaystyle \implies x^2+y^2=2$ $\displaystyle \implies xy=2$ $\displaystyle (y+2)^2+y^2=2$ $\displaystyle 2y^2+4y+2=0\implies y=\frac{4\pm0}{4}$ $\displaystyle x^2+(x2)^2=2$ $\displaystyle 2x^24x+2=0\implies x=\frac{4\pm0}{4}$ Looks like skipjack's guess is correct. Last edited by greg1313; November 10th, 2018 at 06:44 PM. 

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