# tangent of hyperbola

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

 $\displaystyle\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\;=\;1$ (1)

having  $(x_{0},\,y_{0})$  as the tangency point ($y_{0}\neq 0$).

If  $(x_{1},\,y_{1})$  is another point of the hyperbola ($x_{1}\neq x_{0}$), the secant line through both points is

 $\displaystyle y\!-\!y_{0}\;=\;\frac{y_{1}\!-\!y_{0}}{x_{1}\!-\!x_{0}}(x\!-\!x_% {0}).$ (2)

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

 $0\;=\;1\!-\!1\;=\;\left(\frac{x_{1}^{2}}{a^{2}}-\frac{y_{1}^{2}}{b^{2}}\right)% -\left(\frac{x_{0}^{2}}{a^{2}}-\frac{y_{0}^{2}}{b^{2}}\right)\;=\;\frac{(x_{1}% \!-\!x_{0})(x_{1}\!+\!x_{0})}{a^{2}}-\frac{(y_{1}\!-\!y_{0})(y_{1}\!+\!y_{0})}% {b^{2}},$

which implies the proportion equation

 $\frac{y_{1}\!-\!y_{0}}{x_{1}\!-\!x_{0}}\;=\;\frac{b^{2}(x_{1}\!+\!x_{0})}{a^{2% }(y_{1}\!+\!y_{0})}.$

Thus the equation (2) may be written

 $\displaystyle y\!-\!y_{0}\;=\;\frac{b^{2}(x_{1}\!+\!x_{0})}{a^{2}(y_{1}\!+\!y_% {0})}(x\!-\!x_{0}).$ (3)

When we let here  $x_{1}\to x_{0},\;\,y_{1}\to y_{0}$,  this changes to the equation of the tangent:

 $\displaystyle y\!-\!y_{0}\;=\;\frac{b^{2}x_{0}}{a^{2}y_{0}}(x\!-\!x_{0}).$ (4)

A little simplification allows to write it as

 $\frac{x_{0}x}{a^{2}}-\frac{y_{0}y}{b^{2}}\;=\;\frac{x_{0}^{2}}{a^{2}}-\frac{y_% {0}^{2}}{b^{2}},$

i.e.

 $\displaystyle\frac{x_{0}x}{a^{2}}-\frac{y_{0}y}{b^{2}}\;=\;1.$ (5)

Limiting position of tangent

Putting first  $y:=0$  and then  $x:=0$  into (5) one obtains the values

 $x\;=\;\frac{a^{2}}{x_{0}}\quad\mbox{and}\quad y\;=\;-\frac{b^{2}}{y_{0}}$

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 ($x_{0}\to\infty,\;y_{0}\to\infty$), both intersection points tend to the origin.  At the same time, the slope $\frac{b^{2}x_{0}}{a^{2}y_{0}}$ tends to a certain limit $\frac{b}{a}$, because

 $\frac{y_{0}}{x_{0}}\;=\;\frac{b}{a}\sqrt{x_{0}^{2}\!-\!a^{2}}:x_{0}\;=\;\frac{% b}{a}\sqrt{1\!-\!\frac{a^{2}}{x_{0}^{2}}}\;\longrightarrow\,\frac{b}{a}.$

Thus one infers that the limiting position of the tangent line is the asymptote (http://planetmath.org/Hyperbola2)  $y=\frac{b}{a}x$  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  $(x_{0},\,y_{0})$.

Title tangent of hyperbola TangentOfHyperbola 2013-03-22 19:10:31 2013-03-22 19:10:31 pahio (2872) pahio (2872) 12 pahio (2872) Derivation msc 53A04 msc 51N20 msc 51-00 Slope TangentOfConicSection