Relation between quadric and dual quadric

When ${\tt M}$ varies along the quadric, it satisfies ${\tt M}^\top {\bf Q} {\tt M}$ and thus the tangent plane ${\tt\Pi }$ to ${\bf Q}$ at ${\tt M}$ satisfies ${\tt\Pi}^\top {\bf Q}^{-1} {\tt\Pi} = 0$. This shows that the tangent planes to a quadric ${\bf Q}$ are belonging to a dual quadric ${\bf Q}^* \sim {\bf Q}^{-1}$ (assuming ${\bf Q}$ is of full rank).