  Let $F$ be a non-singular homogeneous polynomial of degree $d$ in $n$
variables. We give an asymptotic formula of the pairs of integer points
$(\mathbf x, \mathbf y)$ with $|\mathbf x| \le X$ and $|\mathbf y| \le Y$ which
generate a line lying in the hypersurface defined by $F$, provided that $n >
2^{d-1}d^4(d+1)(d+2)$. In particular, by restricting to Zariski-open subsets we
are able to avoid imposing any conditions on the relative sizes of $X$ and $Y$.
