Why does this graph intercept both axes at the golden ratio?

Question

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.

Answer 1

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.

Answer 2

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.