Find Hyperbola equation from non orthogonal asymptotes

Question

I am looking for an easy way to find the hyperbola that has two non vertical asymptotes $y=m_1x+q_1$ and $y=m_2x+q_2$ and with a vertex located at a distance $r$ from the point where the two asymptotes join.

Would appreciate any help.

Answer 1

Every hyperbola with the given lines as asymptotes has equation of the form $$ \bigl[y - (m_{1} x + q_{1})\bigr]\bigl[y - (m_{2} x + q_{2})\bigr] = c $$ for some real number $c$. (If $c = 0$ you recover the asymptotes.)

If you have a convenient way of getting the coordinates of a vertex (or any point on the hyperbola), evaluating the left-hand side gives $c$. (There are two pairs of potential vertices at distance $r$ from the point where the asymptotes intersect, lying along the lines bisecting the angles between the asymptotes.)