tangent of hyperbola

Let us derive the equation of the tangent line of the hyperbolaPlanetmathPlanetmath

x2a2-y2b2= 1 (1)

having  (x0,y0)  as the tangency point (y00).

If  (x1,y1)  is another point of the hyperbola (x1x0), the secant lineMathworldPlanetmath through both points is

y-y0=y1-y0x1-x0(x-x0). (2)

Since both points satisfy the equation (1) of the hyperbola, we have

0= 1-1=(x12a2-y12b2)-(x02a2-y02b2)=(x1-x0)(x1+x0)a2-(y1-y0)(y1+y0)b2,

which implies the proportion equation


Thus the equation (2) may be written

y-y0=b2(x1+x0)a2(y1+y0)(x-x0). (3)

When we let here  x1x0,y1y0,  this changes to the equation of the tangent:

y-y0=b2x0a2y0(x-x0). (4)

A little simplification allows to write it as



x0xa2-y0yb2= 1. (5)

Limiting position of tangent

Putting first  y:=0  and then  x:=0  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 (x0,y0), both intersection points tend to the origin.  At the same time, the slope b2x0a2y0 tends to a certain limit ba, because


Thus one infers that the limiting position of the tangent line is the asymptote (http://planetmath.org/Hyperbola2)  y=bax  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  (x0,y0).

Title tangent of hyperbola
Canonical name TangentOfHyperbola
Date of creation 2013-03-22 19:10:31
Last modified on 2013-03-22 19:10:31
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Derivation
Classification msc 53A04
Classification msc 51N20
Classification msc 51-00
Related topic Slope
Related topic TangentOfConicSection