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ellipse
1656 (implied in ellpitical), from Fr. ellipse,
from L. ellipsis "ellipse," also, "a falling short, deficit,"
from Gk. elleipsis (see ellipse), because the conic section of
the cutting plane makes a smaller angle with the base than does
the side of the cone, hence, a "falling short." First applied by
Apollonius of Perga (3c. B.C.E.).
ttp://www.etymonline.com/index.php?term=ellipse

hyperbola
1668, from Gk. hyperbole "extravagance,"
lit. "a throwing beyond" (see hyperbole).
Perhaps so called because the inclination of the plane to the
base of the cone exceeds that of the side of the cone.
ttp://www.etymonline.com/index.php?search=hyperbola&searchmode=none