<s>[INST] Find all integers $(x,y)$ satisfying $(x+y+11)^2 = x^2 + y^2 + 11^2$ Find all integers $(x,y)$ satisfying $(x+y+11)^2 = x^2 + y^2 + 11^2$.
So far,
\begin{align*}
x^2 + y^2 + 121 + 2xy + 22x + 22y &= x^2 + y^2 + 121\\
2xy + 22x + 22y &= 0\\
(2x+22)y &= -22x\\
(x+11)y &= -11x
\end{align*}
At least 1 of $x,y$ must be a multiple of 11?
Dont know where to progress after this. All help appreciated.
 [/INST] Hint for another way:Very useful the "Simon's trick" cited above by @John Omielan to solve this kind of diophantine equations.We want to give another kind of solution.
From $11=\dfrac{-xy}{x+y}$ we have $-xy=11K$ and $x+y=K$ so we can put $y=11M$ from which $$-xM=K\text { and }x+11M=K\Rightarrow \begin{cases}x=\dfrac{-11M}{M+1}\\y=11M\end{cases}$$  a parametrization of the variables from which immediately the solutions corresponding to $M=0$ and $M=10$ the other being less immediate but not hard to find.
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