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The eccentricity and directrix of an ellipse. Consider the focal radii of an ellipse

; .

By definition, we have


Consider the difference of squares



or .


d1 (;) d2

r1 r2

F1(c;0) 0 F2(c;0) x


x=l x=l


to determine the focal radii, we solve the system of equations


Definition. The ratio of distances between the foci to the sum of focal radii is called eccentricity:


If the distance between the foci is less than 2, then the eccentricity is

Thus, the focal radii of the ellipse are



Definition. The directrix of an ellipse is the straight line parallel to the y-axis such that the ratio of the focal radius to the distance from an ellipse point to it is constant and equal the eccentricity.

Let us draw two straight lines x=l and x=l parallel to the y-axis and find l such that the ratio of the focal radius to the distance from a point to this straight line is constant and equals the eccentricity:


Substituting the distance and the focal radius, we obtain


The ratio is equal to the eccentricity when , i.e., is the directrix. By analogy, we obtain equations of the directrices:

; ,

where .


Date: 2015-01-02; view: 1821

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