A hyperbola is intersected by line 1 at and

and by line 2 at and

Lines 1 and 2 intersect at

These can be written

I am trying to find a general solution for the values of and .

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