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.