History of Conic Sections Recently, I came to know that ancient Greeks had already studied conic sections. I find myself wondering if they knew about things like directrix or eccentricity. (I mean familiar with these concepts in the spirit not in terminology).
This is just the appetizer. What I really want to understand is what will make someone even think of these (let me use the word) contrived constructions for conic sections. 
I mean let us pretend for a while that we are living in the $200$ BC period. What will motivate  the mathematicians of our time (which is $200$ BC) study the properties of figures that are obtained on cutting a cone at different angles to its vertical? 
Also, what will lead them to deduce that if the angle of cut is acute then the figure obtained has the curious property that for any point on that figure the sum of its distances from some $2$ fixed points a constant.
And in the grand scheme of things how do our friend mathematicians deduce the concepts of directrix and eccentricity (I am not sure if this was an ancient discovery, but in all, yes I will find it really gratifying to understand the origin of conic sections). 
Please shed some light on this whenever convenient. I will really find it helpful.
Thanks
 A: As others noted, Apollonius wrote 8 volumes on conics.
For further info, try this web site or this book by Coolidge.
One very early motivation was apparently the design of mirrors for burning things.
A: I'm not a historian.  However, my guess is that ellipses, parabolas, and hyperbolas were initially defined as certain loci, and then only later discovered to be cross-sections of a cone.  At least, if I were living back in 200 BC (as you say), this is how I would've gone about it.
Let's be slightly more concrete.  I hope you can agree that circles are intrinsically important.  Now, a circle is really just the set of points at a fixed distance from a given fixed point (the center).  So, as ancient geometers, we should use this "locus" idea to find other important shapes.
So, we are led to think about the set of points which are equidistant between a fixed point and a fixed line.  This turns out to be a parabola.  (The fixed line we chose is called the "directrix.")  Or, we could ask about the set of points whose distances from two fixed points add up to a given fixed number: this is an ellipse.  And if we replace "add up to" with "subtract to," we get a hyperbola.
To me, these definitions are simple and natural.  The truly amazing thing is that every one of these constructions can be described as a cross section of a cone.  Wow!
(Again, though, I should emphasize that I'm really just guessing how these shapes were initially constructed.  So where the actual history is concerned, I could be totally wrong.)
A: The initial motivation seems to have been the solution of the Duplication of the Cube which is with the Quadrature of the Circle one of the most famous problems of Antiquity.
See Sir Thomas Heath's modern adaptation of Apollonius' Treatise on Conic Sections with a detailed history of the matters concerning you.
It can be downloaded freely from
http://www.wilbourhall.org/pdfs/Treatise_on_Conic_Sections.pdf
Dr R J-M Grognard
A: There are several Ideas where it might have come from.
One such idea is the construction of burning mirrors, for which a parabola is the best shape, because it concentrates the light in a single point, and the distance between the mirror and the point can be calculated by the use of geometry (see diocles "on burning mirrors", I could really recommend this book for it's introduction alone).
Conics where also usefull in the construction of diferent sun dials.
I have researched the topic quite a bit but sadly I am yet to understand how did they "merge" all this seemingly unrealted topics to cutting a cone.
Most likely the lost work "on solid loci" from euclid would provide some more insight.
