tangent line at point p on hyperbola y=c/x

[treeman]

Let c be a positive real number and P be a point on the component of hyperbola y=c/x lying in the first quadrant. Consider the tangent line to the hyperbola y=c/x at the point P.

1. The tangent line meets the x-axis at a point A and the y-axis at the point B. Show that P is the midpoint of the segment AB

2. Show that the triangle ABO, where O is the origin, has the same area, no matter where P is located on the hyperbola.

Noone in my math class that i know of got this problem correct.
your help is greatly appreciated :-D