If X is a hypersurface of degree d in $\mathbb{P}^n$ and S is the singular locus of X then when is it true that the secant line of two points in S is contained in X? I think this has something to do with intersection theory but I really have no idea about how to think about this question. For cubic hypersurfaces I've seen that the secant line connecting two singular point of the hypersurface lies in the hypersurface but is this true for higher degree hypersurfaces?