We give a classification of generalized conics analogous to the classical ones due to the weights and locations of the focal points. We also examine generalized conics with respect to several properties that hold for classical conics. For various mechanical and optical properties of classical conics, we show which of them apply to generalized conics. From this treatment we get another generalization of conics in the plane. The results in this article are also interesting in connection with the problem of determining the Fermat-Torricelli points. We construct examples of configurations of focal points in certain norms where the Fermat-Torricelli set contains points that do not lie in the convex hull of the focal points.
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