How to construct a tangent to a hyperbola that is parallel to a given line?

Question

You are given a hyperbola $h$, its asymptotes and its foci. You are also given some line $p$. Construct the line(s) tangent to $h$ and parallel to $p$.

This problem came up while I was doing something and I can't solve it. Google doesn't help much either. Any help is appreciated.

Answer 1

1. Construct the center $S$ of the hyperbola (it's just the intersection of its asymptotes).
2. Draw a parallel $p'$ to $p$ which passes through $S$. The construction is possible only if $p'$ doesn't intersect $h$, otherwise, it is impossible.
3. Take any point $P$ on $p'$ other than $S$
4. Draw parallels to the asymptotes through $S$. Denote their intersections with the asymptotes by $A$ and $B$.
5. Draw the line $l$ parallel to $AB$ through $S$
6. Denote the intersections of $l$ with $h$ by $C$ and $D$
   
   

And that's it. The tangents at $C$ and $D$ will be parallel to $p$. To see that just note that $S(A,B;P,C)$ is a harmonic quadruple of lines and therefore the polar of $l_{\infty}=l'_{\infty}$ has to be $CD$. One can also prove the result by considering an affine transformation taking $p'$ to the axis of symmetry of the hyperbola (the one that does not intersect it).

Answer 2

A parallel line will have the same slope as p. So you can solve the derivative of the hyperbola for which x its derivative is the same as p, and then find the y-value of the line parabola at that point.