I apologise if this question is a duplicate, I couldn't find it anywhere.
I came across the following problem:
An ordered pair $(x,y)$ of integers is called a primitive point if the greatest common divisor of $x$ and $y$ is $1$.Given a finite set $S$ of primitive points, prove that there exists a positive integer $n$ and integers $a_0, a_1,\ldots, a_n$ such that for all $(x,y) \in S$ we have that $$a_0x^n\ + \ a_1x^{n-1}y\ +\ a_2x^{n-2}y^2\ + \ldots+\ a_{n-1}xy^{n-1} \ + \ a_ny^n=1.$$
How does one go about proving this?
What first came to mind was to use induction on this problem. I believe this should work for any points $(x,y)$ where $x$ and $y$ are relatively prime. For the base case $n=1$ we would have that $$ax +by=1$$ for some integers $a$ and $b$ which we know exist because of Bézout's identity. However I don't know how to follow up or even if induction is the proper way to tackle this problem. It also seems to me this has a strange similarity to the binomial theorem, could they be connected somehow?
Any hints and/or ideas are welcome!