tangent of hyperbola

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

${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=\mathrm{\hspace{0.33em}1}$(1)

having  $(x_0,y_0)$  as the tangency point ($y_0\ne 0$).

If  $(x_1,y_1)$  is another point of the hyperbola ($x_1\ne x_0$), the secant line through both points is

$y-y_0={\displaystyle \frac{y_1-y_0}{x_1-x_0}}(x-x_0).$(2)

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

$$0=\mathrm{\hspace{0.33em}1}-1=\left(\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}\right)-\left(\frac{x_0^2}{a^2}-\frac{y_0^2}{b^2}\right)=\frac{(x_1-x_0)(x_1+x_0)}{a^2}-\frac{(y_1-y_0)(y_1+y_0)}{b^2},$$

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

$y-y_0={\displaystyle \frac{b^2(x_1+x_0)}{a^2(y_1+y_0)}}(x-x_0).$(3)

When we let here  $x_1\to x_0,y_1\to y_0$,  this changes to the equation of the tangent:

$y-y_0={\displaystyle \frac{b^2{x}_0}{a^2{y}_0}}(x-x_0).$(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.

${\displaystyle \frac{x_{0}x}{a^2}}-{\displaystyle \frac{y_{0}y}{b^2}}=\mathrm{\hspace{0.33em}1}.$(5)

Limiting position of tangent

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

$$x=\frac{a^2}{x_0}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}y=-\frac{b^2}{y_0}$$

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 ($x_0\to \mathrm{\infty },y_0\to \mathrm{\infty }$), 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

$$\frac{y_0}{x_0}=\frac{b}{a}\sqrt{x_0^2-a^2}:x_0=\frac{b}{a}\sqrt{1-\frac{a^2}{x_0^2}}⟶\frac{b}{a}.$$

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  $(x_0,y_0)$.