Tangent to hyperbola without using derivative

Question

I'm asked to calculate the tangent to a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at a point $P(x_1,y_1)$ without using derivatives (or limits), and we cannot use geometric transformations, because they 're not linear when applied to non-bounded curves (this is also something I would like an explanation to).

Thanks in advance!

Answer 1

Hint.

You can use the equation of the lines thorough $P$ and take the system with the equation of the hyperbola. $$ \begin{cases}y-y_1=m(x-x_1)\\ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \end{cases} $$

the value of $m$ such that this system has only one solution it the slope of the tangent. (do you see why?)

If $P$ is a point of the hyperbola you find one only value of $m$.

Do you know how to find this value of $m$?

Answer 2

Use an optical property of the hyperbola: a ray thrown from on focus to $(x1, y1)$ reflects as a continuation of the ray thrown from another focus. The bisector of the angle between the rays is normal to hyperbola.

The foci of your hyperbola are $(0, \pm\sqrt{a^2+b^2})$.