Assume that $Q$ is a proper (not affine subspace) hypersurface in $\mathbb R^n$ of the second degree given by equation $ \sum_{i,j=1}^n a_{ij}x_i x_j +2\sum_{i=1}^n b_i x_i +c=0, $ where $a_{ij}=a_{ji}$; $a_{ij}, b_i,c\in \mathbb R$.
Is it true that if $Q$ is symmetric with respect to a hyperplane $p+H$, where $H$ is a $n-1$ dimensional hyperlane with $0 \in H$, $p\in \mathbb R^n$, then the asymptotic cone $C$ of $Q$:
$ \sum_{i,j=1}^n a_{ij}x_i x_j=0 $
is symmetric with respect to $H$ ?