Dionysodorus of Amisus in Pontus applied the intersection of a parabola and hyperbola to the solution of a problem which Archimedes, in his Sphere and Cylinder , had left incomplete. The problem is "to cut a sphere so that its segments shall be in a given ratio."
Archimedes studied also the ellipse and accomplished its quadrature, but to the hyperbola he seems to have paid less attention. It is believed that he wrote a book on conic sections.Florian Cajori
It is worthy of notice that Apollonius nowhere introduces the notion of directrix for a conic, and that, though he incidentally discovered the focus of an ellipse and hyperbola, he did not discover the focus of a parabola. Conspicuous in his geometry is also the absence of technical terms and symbols, which renders the proofs long and cumbrous.Florian Cajori
By the word 'conoid,' in his book on Conoids and Spheroids, is meant the solid produced by the revolution of a parabola or a hyperbola about its axis. Spheroids are produced by the revolution of an ellipse, and are long or flat, according as the ellipse revolves around the major or minor axis. The book leads up to the cubature of these solids.Florian Cajori