here is what i would do:
the eigenvector you have for a+b is a bit messy, in the denominator, √(1+(1/b)^2) = √((b/b)^2+(1/b)^2) = √[(1+b^2)/b^2],
so you could have chosen (1/√(1+b^2), b/√(1+b^2)) as the eigenvector instead (which matches the first eigenvector in style, at least).
now, re-calculate det(Q), i think your answer should have a's AND b's in it (when i did it, it was kinda ugly).
you might also wish to rationalize your denominators, by multiplying the first eigenvector by √(1 + a^2)/√(1 + a^2), and similarly for b.