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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 . y

d1 М(х;у) 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  or 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 2а, 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 М 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: 906

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