Let $X$ be an $n$-dimensional differensiable submanifold of euclidean $\mathbb{R}^N$. Let $P_0$ be a point of $\mathbb{R}^N-X$ and denote by $\phi(P)=|P-P_0|$, the distance function from $P_0$ to $P$. The critical points of $\phi$, the points where the vector $P-P_0$ is orthogonal to $X$.
Let $P$ be a critical point for the distance function $\phi$. Choose rectangular coordinates $x_l, ..., x_n, ..., x_N$, in $\mathbb{R}^N$ with origin at $P$ and with the sp $x_{n+1} = ... = x_N = 0$ as tangent space to $X$ at $P$. Then $x_1,..., X_n$ can used as local coordinates in $X$ at $P$. Let the fixed point $P_0$ have coordinates $(0,..., 0, a_1,.., a_{N-n}$).
My question is, why near $P$, $X$ is represented by equations of the form $x_{n+i}=\frac{1}{2}\sum_{i,j=1}^nb_{ij}^ix_ix_j+H^i(x_1,...,x_n)$ for $1\le i\le N-n$? where $H^i$ together with its first and second partial derivatives vanish at $P$.