A line intersects a hyperbola at the points $(-2,-6)$ and $(4,2)$. It also intersects one of the asymptotes of the hyperbola at the point $(1,-2)$.
Find the centre of the hyperbola.
My approach:
The mid-point of the two points of contact $(-2,-6)$ and $(4,2)$ is the point $(1,-2)$ and we are given that the line cuts the asymptote at this point. Hence the answer is $(1,-2)$. But this seems intuitive and I am not sure whether my reasoning is correct. I am looking for another approach to the problem.