Here's where I am so far...
then we need
Is this as far as I can go?
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
I had forgotten about one of the original constraints
This works out to
Taking this with the from the previous result I have
So for integer solutions must be a perfect square.
With the original equations
The only way to find values for is by trial and error.
I am trying to find a general solution.
It is similar to the diophantine solution for a pythagorean triple
In the attempt I've made so far, the
and whatever will work out to be ( a similar expression to y)
If I can make sure that x and y are integers, I would have correct numbers to satisfy the original equations
OK, I'll try giving an example of what I'm looking for
If one sets
By choosing any integer m>1 a, b and c will be a pythagorean triple.
This doesn't give all pythaorean triples, but it does give an infinite amount of them, which is usually enough
I'm looking for a similar way of getting numbers that satisfy the 2 equations in the op.
I had made further progress from my last post after I noticed that
u_2 is not only greater than u_1, but also
This satisfies the first equation. When solving with the second equation I would have
which is no further forward
Quick synopsis of where I am at the moment
Solving for being a perfect square leads me back to the beginning.