Let $P$ be the set of all parabolas with the equation of the form $$y=(a-1)x^2-2(a+2)x+a+1$$where $a$ is a real parameter and $a\neq1$. Prove that there exists an unique point $M$ such that all parabolas in $P$ pass through $M$.
Source: EGMO TST - Moldova 2024 P1
Tags: parabola, algebra, parameterization
Let $P$ be the set of all parabolas with the equation of the form $$y=(a-1)x^2-2(a+2)x+a+1$$where $a$ is a real parameter and $a\neq1$. Prove that there exists an unique point $M$ such that all parabolas in $P$ pass through $M$.