Dual conic
= Dual conic
{title2=$\ell^{\mathsf T}Q^{-1}\ell=0$}
The tangent lines of a <smooth plane conic> form a <smooth plane conic> in the <dual projective space>. If the original equation is $v^{\mathsf T}Qv=0$, its tangent has coefficient vector $Qv$, giving the displayed equation. The matrix $Q$ is symmetric and invertible, so this correspondence is a projective linear isomorphism of the conics.