tangent of hyperbola
having as the tangency point ().
If is another point of the hyperbola (), the secant line through both points is
Since both points satisfy the equation (1) of the hyperbola, we have
which implies the proportion equation
Thus the equation (2) may be written
When we let here , this changes to the equation of the tangent:
A little simplification allows to write it as
Limiting position of tangent
Putting first and then into (5) one obtains the values
on which the tangent line intersects the coordinate axes. From these one sees that when the point of tangency unlimitedly moves away from the origin (), both intersection points tend to the origin. At the same time, the slope tends to a certain limit , because
Thus one infers that the limiting position of the tangent line is the asymptote (http://planetmath.org/Hyperbola2) of the hyperbola.
Consequently, one can say the asymptotes of a hyperbola to be whose tangency points are infinitely far.
The tangent (5) halves the angle between the focal radii of the hyperbola drawn from .
|Title||tangent of hyperbola|
|Date of creation||2013-03-22 19:10:31|
|Last modified on||2013-03-22 19:10:31|
|Last modified by||pahio (2872)|