Let the hyperbole equation be:

**x*y=C**

Let's C=1 to simplify the proof.

The tangent to the curve in point P(x1,y1) will be:

**y-y1=-(1/x1^2)*(x-x1)**

Getting OM & ON is obtained by inserting Y=0 and X=0 respectively.

**We get OM=2*x1**

and ON=2*y1

So the area will be

S= .5*OM*ON=.5*4*X1*Y1=2*X1*Y1=2

So

** S=2 - not related to coordinates of the point P.**

*Edited on ***October 18, 2014, 4:26 am**