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!