Proposition 9.

Proposition 9. The product of the distances from the point M(x,y) to the hyperbola’s asymptotes is constant and is equal to a²b²/(a²+b²)
Supplementary problem
  • Prove that the vertices of the hyperbola and the points of intersection of its asymptotes with directrixes belong to a circle.
  • Prove that the segment of the tangent concluded between the asymptotes of the hyperbola is halved in the point of contact.