We study conic sections because they are heavily used in astronomy because more often than not, the orbit of an object is elliptical in nature.
Also, it has heavy applications in something called the 'slingshot effect':
http://webphysics.iupui.edu/warmup/iupui_archive/152gif/trajectory_lg.gif
Where we utilise a planet's gravity to 'slingshot' the satellite out of the orbit. At what angle and velocity should the satellite travel at, such that it escapes the orbit?
http://en.wikipedia.org/wiki/Gravity_assist
Think of the planet as a kind of 'focus' in which the gravity is concentrated.
Also, more recently appearing in cycling, are elliptical gears as opposed to circular gears, because it decreases chances of jams when changing gears:
http://en.wikipedia.org/wiki/Ellipse#Elliptical_gears
The parabola is a conic section too, and you may know that it has huge applications in physics as a light reflector, a satellite receiver and of course in the study of projectiles.
Hyperbolas have many applications in the study of GPS systems, using the definition of it with the eccentricity (maintaining a fixed ratio) in a process called 'Trilateration':
http://en.wikipedia.org/wiki/Trilateration
Not to mention the circle, which is also a conic section, and I highly doubt I need to go into detail about how much the circle is used in real life.
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