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

[HallsofIvy]

Write the equation as xy= c. Then, differentiating with respect to x, y+ xy'= 0 or y'= -y/x. At the specific point (x0,y0), the slope is -y0/x0 so the tangent line is y= (-y0/x0)(x- x0)+ y0.

At point A, y= 0 so (-y0/x0)(x- x0)= -y0, x- x0= -y0(-x0/y0)= -x0, and x= 2x0.

At point B, x= 0 so y= (-y0/(x0)(-x0)+ y0= 2y0.

That was easy, wasn't it?

Now, what is the area of the triangle ABO?