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Recently I've become familiar with conics as my high school academics include this topic. My textbook has a brief discussion where I learned how circle, ellipse, parabola and hyperbola are defined from sections of a cone. Also, in my textbook I found an algebraic representation which introduces a fixed point $S$, namely 'focus', and a fixed line ('directrix') $l$; as I am taught, a conic is the locus of all points $P$ that satisfies $\frac{PS}{PM}= C$, where $C$ is a constant and $PS$ & $PM$ are the perpendicular distances of the point from $S$ & $l$.

My question is, what is the motivation of the algebraic definition? How can I link up these two definitions together?

I am not looking for proof firsthand, I would like to know the intuitions.

J. W. Tanner
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tanim13
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  • Welcome to Math.SE! ... and to the fascinating world of conics! :) ... This ancient answer of mine may provide some insights about your question. More fundamentally, the notion of Dandelin spheres shows how to interpret the focus and directrix of a conic in relation to the cone and plane that define it. – Blue Nov 13 '23 at 19:26
  • You begin by defining the cone in three dimensions and the plane then find the intersection by discovering the points which satisfy both simultaneously. The eccentricity, $\frac{PS}{PM}$, encodes the slope of the plane relative to the cone. – CyclotomicField Nov 13 '23 at 20:03
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    What is "ig" (in the title)? – mr_e_man Nov 13 '23 at 20:30
  • The definition as locus has the advantage of being restricted to the plane: no need for 3D geometry. – Intelligenti pauca Nov 13 '23 at 21:07
  • It is not correct that the focus/directrix definition is "algebraic". It was known to the ancient Greeks long before algebra was invented. – Paul Sinclair Nov 14 '23 at 16:48

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