Homology of Generalized Hyperbola

Question

Consider $H\le R^{a+b}$ defined by $$H = \{x\mid x_1^2+\cdots+x_a^2-x_{a+1}^2-\cdots-x_{a+b}^2=1\}$$ with some $b\ge 1$. This is a kind of 'generalized parabola' although I find it impossible to concieve in $\ge 4$ dimensions. I am trying to find the homology of this space. My idea was to try and integrate the Mayer–Vietoris sequence with induction on something, $b$ perhaps, to reduce the problem to a simpler one. However even if we take the simplest non-trivial case, $b=1$, this still seems remarkably impregnable.