A line tangent to a hyperbola intersects its asymptotes at points L and N.
Let M be the intersection of the asymptotes.

Prove that the area of ΔLMN is a constant.
  

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

Posted by Ady TZIDON on 2014-10-18 04:16:30