Let $X\subset \mathbb P^n=\mathbb P^n_\mathbb C$ be a smooth hypersurface of degree $d$, $d>1$. For any $x\in X$, the tangent plane $H$ intersects with $X$ at $x$, thus $X\cap H$ is singular at $x$. So, we can look at the tangent cone of $X\cap H$ at $x$. If the the cone is non-degenerate, i.e. the quadratic form is of full rank, then we call $x$ is a good point, otherwise we call a bad point. We denote the set of good (resp. bad) points by $G$ (resp. $B$).
I would like to know
Is there a good way to describe what is $B$?
In particular, I suppose the following statement is always true:
The set $B$ is a proper (Zariski) closed subset of $X$.
But I did not find how to prove it.
Also, I would like to know how bad a point can be. In an affine chart, we can express it as the zero locus (denoted by $Z(f)$) of $f=f_0+f_1+f_2+\ldots$, where $f_i$ is the degree $i$ part. $f_0(x)=0$ means $x\in Z(f)$; $f_0(x)=f_1(x)=0$ means $x$ is a singular point of $Z(f)$. If $f_i(x)=0$ for all $i\leq k$, we call $x$ is singular to the $k$-order. If $x\in X$ is singular point of $X\cap H$ and singular to the $k$-order, we denote $x\in B_k$. Easy to see $$X=B_1 \supset B\supset B_2\supset B_3 \supset \ldots\supset B_d=\emptyset.$$
I want to know that
Is there some $k$ such that $B_k=\emptyset$ for all $(n,d)$ and a generic $X$?
Thanks in advance.