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.
  

Could we take Ady’s rectangular hyperbola and stretch it in the
two directions at 45 degrees to the asymptotes to such an extent that,
while its centre remains fixed, the curve now coincides with any
hyperbola we choose. The area of any region in the plane will change by the same proportion depending only on the stretch factors, so all Ady’s triangles with equal areas will have new areas but they will all be equal wherever the tangent is drawn.

Edited on October 19, 2014, 7:04 pm

Posted by Harry on 2014-10-19 19:01:45