First of all, we may consider the asymptotes as two specific tangents :
the contact point is at infinity.
Apart this, other general properties of tangents apply as well to asymptotes. For instance :
Perpendicular projection Q of a focus on an asymptote lies on the principal circle.
Hence OQ = a
The product of distances from focii to an asymptote equals b²
FQ.F'Q' = b², that is FQ = F'Q' = b
The tangent in apex hhas length 2b (between the les asymptotes) :
AB = FQ
Equation of hyperbola from its axis x²/a² - y²/b² = 1
a = OA = semi axis, c = OF = OF', a² + b² = c²
In an analytic way, the asymptote directions of the conic section
ax² + bxy + cy² + dx + ey + f = 0 are obtained by solving
a + bm + cm² = 0 (in homogenous coordinates, Z=0).
And they intersect at center of the conic section.
If the coordinate system axes are the asymptotes, the hyperbola equation becomes very simple : xy = k
When the asymptotes are perpendicular, the hyperbola is said "equilateral".
MN and PQ have same midpoint
This can be proved easily from the hyperbola equation :
Let M and N the intersect points of line x/a + y/b = 1 and hyperbola xy = k.
The abscisses of M and N are then solutions of x²/a + k/b = x, that is x² - ax + ak/b = 0.
The sum of roots being a, midpoint of MN has abscisses a/2, which is the same as the midpoint I of PQ : P has coordinates (a,0) and Q = (0,b).
The touch point is the midpoint of PQ
|OPQIJ are on a same circle.|
Axes being the angle bisectors of asymptotes intersect the perpendicular bisector of PQ on the circumcircle to OPQ.
Hyperbola is the envelope of secants PQ with Area(OPQ) = constant