A hyperbola is intersected by line 1 at $\displaystyle (u_1, v_1)$ and $\displaystyle (v_1, u_1)$

and by line 2 at $\displaystyle (u_2, v_2)$ and $\displaystyle (-v_2, -u_2)$

Lines 1 and 2 intersect at $\displaystyle (u_1+v_1, 0)$

These can be written

$\displaystyle u_1+v_1=u_2-v_2\ \ \ \ \ \ \(1)$

$\displaystyle u_1v_1=u_2v_2\ \ \ \ \ \ \ \ \ \ \ \ \ (2)$

I am trying to find a general solution for the values of $\displaystyle u_1, v_1, u_2$ and $\displaystyle v_2$.

I have tried squaring (1) and adding\subtracting various multiples of (2), but this just brings me various fractions of the starting values.

Any help or pointers would be greatly appreciated