Can you please help me solve this system of equations (frankly I have no idea, it's the first equation of this type that I solve, so please, write only a hint):

$$ \left\{ \begin{array}{c} x-\arctan y=y-\arctan x \\ x^2-xy+4y^2=4 \end{array} \right. $$

Thank you!!


1 Answer 1


Without loss of generality we may suppose that $x\ge y$. But if $x>y$, what is the sign of $(\arctan y-\arctan x)$ ? conclude that $x=y$ and finish the job.

  • $\begingroup$ But $x=y$ won't satisfy the second equation... $\endgroup$ Apr 28, 2014 at 17:53
  • $\begingroup$ @Paul Safier: Sure? $x=y=1$ is a solution. $x=y$ must not "satisfy", it must be substituted into the second equation. $\endgroup$ Apr 28, 2014 at 17:56
  • $\begingroup$ $x=y=1$ is the only solution, according to Omran. $\endgroup$ Apr 28, 2014 at 17:59
  • $\begingroup$ Agreed. I suppose $x=y=-1$ might also be a solution if that's an accepted portion of the domain... $\endgroup$ Apr 28, 2014 at 18:15

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