  For each adjoint variety not of type $A$ or $C$, we study the irreducible
component of the Hilbert scheme which parametrizes all smooth conics. We prove
that its normalization is a spherical variety by using contact geometry, and
then compute the colored fan of the normalization. As a corollary, we describe
the conjugacy classes of conics in the adjoint variety and show smoothness of
the normalization. Similar results on the Chow scheme of the adjoint variety
are also presented.
