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: 20,481 Thanks: 2041 
At a guess, 1.

November 10th, 2018, 05:10 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,932 Thanks: 1127 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 05:44 PM. 

Tags 
number, points, rational 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
rational number  Roli  Algebra  14  June 10th, 2014 01:40 AM 
Critical Points of a Rational Function  Tzad  Calculus  8  November 7th, 2013 07:55 PM 
Rational number  tva_vlad  Algebra  1  October 7th, 2013 01:38 AM 
Centers of a triangle with rational points  Rejjy  Algebra  2  December 1st, 2012 02:41 AM 
Rational Points on a circle  guru123  Algebra  18  June 22nd, 2012 01:38 PM 