Well, you sure got me confused! We start with:
(j*b) / (1 - j) = 2*j*(a - j)
We're to solve for a and b;
b is straightforward: b = [2j(a - j)(1 - j)] / j
Solving for a leads to:
a = [2j(j - 1) - b] / [2(j -1)]
That checks out perfectly with the initial equation.
And there's loads of solutions; examples with all positive integers:
(a,b,j) = (1,2,2), (2,4,3), (1,8,3), (3,6,4), (4,8,5)
Can you explain why you jump to j^2 = -1 ?