This question has been discussed on Math overflow:
This is about conic sections in general. In my answer I gave them as example
of "useless mathematics which became useful centuries later". Some tried to challenge this statement.
One of the conjectures mentioned that conic sections were discovered when the Greeks tried to develop a theory of sundials.
(Normally you get a hyperbola rather than ellipse, an ellipse will arise only in polar regions on the days when Sun never sets). This conjecture was much discussed in the history literature but on my opinion this is an unlikely explanation how conic sections were discovered. Another conjecture says that Menachem used conic sections for doubling the cube. This conjecture is nore plausible. See the references in the link above.
EDIT. There is of course a well-known legend about Archimedes burning Roman ships with parabolic mirrors, and it is clear that this is only a legend: no serious scientist believes that this can be actually performed. (But you can see this in the movies).