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Ellipse


Whereas in a , two distinct points, F1and F2 and 2a being a real number greater than the distance between F1 and F2, we call Ellipse the set of the points of the plane such that the sum of the distances of these points to F1 and F2 always be equal to 2nd.

For example, being P, Q, R, s, F1 and F2 points of the same plane and F1F2 <2a, we have:


The figure obtained is an ellipse. Comments:

1st) The Earth describes an elliptical path around the sun, which is one of the foci of this path. The moon around the earth and the other satellites relative to their respective planets also exhibit this behavior.

2) Halley's comet follows an elliptical orbit, with the Sun as one of its focuses.

3) Ellipses are called conical because they are configured by cutting into a straight circular cone through an oblique plane in relation to its base.

Elements

Notice the following ellipse. In it, we consider the following elements:

  • spotlights: The dots F1 and F2

  • center: the point O, which is the midpoint of

  • larger half shaft: The

  • minor half shaft: B

  • focal half distance: ç

  • vertices: The dots THE1, THE2, B1, B2

  • major axis:

  • minor axis:

  • focal distance:

Fundamental relationship

In the figure above, applying Pythagoras' theorem to the OF triangle2B2 , rectangle in O, we can write the following fundamental relation:

The2 = b2 + c2

Eccentricity

We call it eccentricity the real number and such that:

By the definition of ellipse, 2c <2a, then c <a and, consequently, 0 <and <1.

Note: When the foci are too close, ie c is too small, the ellipse approaches a circle.

Next: Ellipse Equations