# Intersection of two hyperbolas

1. Sep 22, 2008

### uocan

Consider two hyperbolas on t1xt2 plane given below. I want to test whether the two intersect or not. Actually I am trying to prove that the two intersect over a countable set of parameters, a1, b1 etc, thus I don't want to jointly solve for the two equations (doing so results in a second order polynomial equation, and for the curves to intersect the discriminant needs to be positive but testing this over the set of parameters is difficult). Note that the asymptotes of the hyperbolas have the same slope. So my question is, whether there is an easy criterion for intersection of hyperbolae or not? Also, is there a theorem giving sufficient conditions for intersection of hyperbolae with asymptotes of same slope?

\begin{align} ( t_1+ \frac{b_1}{a_1} t_2+\frac{c_1}{a_1})( t_1+\frac{b_3}{a_3} t_2 +\frac{c_3}{a_3})= \frac{a_5}{\pi_i a_1 a_3} t_1 +\frac{c_5}{a_1 a_3} \\ ( t_1+ \frac{b_1}{a_1} t_2+\frac{c_2}{a_2})( t_1+\frac{b_3}{a_3} t_2 +\frac{c_4}{a_4})= \frac{b_6}{\pi_j a_2 a_4} t_2 +\frac{c_6}{ a_2 a_4} \end{align}