The first person to discover the conic section curves was Menaechmus. He was born in 380 BC and was a friend of Plato. (1) He discovered these curves by "examining the intersection of 'a right circular cone of a varying vertex angle' and 'a plane perpendicular to [an] element of the cone'". (3)
Another mathematician that studied conic sections was Archimedes. Archimedes was a great mathematician who worked in many branches of mathematics. He published a few books that mentioned conics, but he never published a book specifically on conics. He defined what each of the conic sections were and defined the names we use today.
Apollonius continued the work of Archimedes with conic sections. He was also known as "The Great Geometer" and studied the properties of conic sections. He wrote a series of books called Conic Sections about the various conic sections. (4)
Hypatia of Alexandria was "the first notable woman mathematician" and she contributed a lot to mathematics. She was the daughter of Theon, who was a professor at the University of Alexandria. He tutored her in math, astronomy and philosophy. It was then that she started her work in mathematics. Later in life, she lectured about conic sections. She commented on Apollonius' work in On the Conics of Apollonius. She helped make these concepts easier to understand through her work.