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Earlier, I was playing around with the Desmos Graphing Calculator, and I discovered that the following formula intercepts both the x and y axes at the golden ratio. I know that it makes sense, but I would like to know if there is any sort of reason.

Graph: $x^2+y^2=(x+1)(y+1)$

Thank you.

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    $\begingroup$ At each axis you have either $x=0$ or $y=0$. The solution to the remaining equation is always the golden ratio. $\endgroup$
    – abiessu
    Dec 12, 2014 at 0:35
  • $\begingroup$ @abiessu I had already confirmed this, but I was wondering if it were due to any other mathematical reason. Below, Meelo has posted an answer. $\endgroup$
    – capturographer
    Dec 12, 2014 at 0:37
  • $\begingroup$ You can see by symmetry that the $x$- and $y$-intercepts are the same. $\endgroup$
    – augurar
    Dec 12, 2014 at 0:57

2 Answers 2

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Algebraically, if we set $y=0$, then this becomes $$x^2=x+1$$ which is the quadratic polynomial of which the golden ratio is a root. Generally, wherever the golden ratio appears, it's because this polynomial showed up.

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This is precisely because the golden ratio is a solution to $$t+1=t^2.\tag{$\star$}$$ In the case that $x=0$ or $y=0$ (but not both), we have that whichever of the two is non-zero may be the golden ratio, or may be the other solution to $(\star),$ but cannot be anything else.

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