  In noncommutative algebraic geometry, noncommutative quadric hypersurfaces
are major objects of study. In this paper, we focus on studying noncommutative
conics $\operatorname{Proj_{nc}} A$ embedded into Calabi-Yau quantum projective
planes. In particular, we give complete classifications of homogeneous
coordinate algebras $A$ of noncommutative conics up to isomorphism of graded
algebras, and of noncommutive conics $\operatorname{Proj_{nc}} A$ up to
isomorphism of noncommutative schemes.
