Directrix, conic section

Conic Sections The cui ves included in this group are obtained from plane sections of the cone. They include the circle, ehipse, parabola, hyperbola, and degeneratively the point and straight line. A conic is the locus of a point whose distance from a fixed point called the focus is in a constant ratio to its distance from a fixea line, called the directrix. This ratio is the eccentricity e. lie = 0, the conic is a circle if 0 < e < 1, the conic is an ellipse e = 1, the conic is a parabola ... 

A conic section or cotdc is the locus of a point which moves so that its distance from a fixed point (called the focus) is in a constant ratio (called the eccentricity) to the distance from a fixed straight line (called the directrix). 

This reflects the fact that all parabolas have the same eccentricity, namely 1. The eccentricity of a conic section is the ratio of the distances point-to-focus divided by point-to-directrix, which is the same for all the points on the conic section. Since, for a parabola, these two distances are always equal, their ratio is always 1. 

Directrix—The fixed line in the focus directrix definition of a conic section. 

The equations for conic sections can be expressed rather elegantly in polar coordinates. As shown in Fig. 5.9, the origin is defined as the focus and a line corresponding to x = d serves as the directrix. Recall the relations between the Cartesian and polar coordinates x = rcos 9, y = rsin6>. The point Fwill trace out the conic section, moving in such a way that the ratio of its distance to the focus r to its distance to the directrix d — x = d—rcos9 is a constant. This ratio is called the eccentricity, e (not to be confused with Euler s 6 = 2.718...) ... 

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