Let the point of tangency be $\left(p,\dfrac{a}{p}\right)$
Tangent line at that point is ...
$y=-\dfrac{ax}{p^2}+\dfrac{2a}{p}$
Point B is $\left(0,\dfrac{2a}{p}\right)$
Point A is $\left(2p,0\right)$
Midpoint of AB is $\left(p,\dfrac{a}{p}\right)$
conclusion is ...
Your last question has been moved to a new thread. Multiple problems in the same thread tends toward confusion. In future start new problems with a new thread. Thank you.