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Ellipse

 

Ellipse. Focuses. Equation of ellipse. Focal length.  

Large and small axes of ellipse. Eccentricity.

Equation of tangent line to ellipse.

Tangency condition of straight line and ellipse.

 

An ellipse ( Fig.1 ) is called a locus of points, a sum of distances from which to the two given points  F1  and  F2 , called  focuses of ellipse, is a constant value.

                                              

An equation of ellipse ( Fig.1 ) is :

                                                                             

Here the origin of coordinates is a center of symmetry of ellipse, and the coordinate axes are its axes of symmetry. At a > b  focuses of ellipse are placed on axis  ( Fig.1 ), at  a < b  focuses of ellipse are placed on axis  Y , and at  a = b  an ellipse becomes a circle ( in this case focuses of ellipse coincide with a center of circle ). Thus, a circle is a particular case of an ellipse. The segment  F1F2 = 2 , where   is called  a focal length. The segment  AB = 2 a  is called  a large axis of ellipse, the segment  CD = 2 b  is called  a small axis of ellipse. The number  e = c / ae < 1 is called an eccentricity of ellipse.

 

Let   ( 1 ,  1 ) be a point of ellipse, then  an equation of tangent line to ellipse  in this point is

                                                                             

A tangency condition of a straight line  y = m x + k  and an ellipse   2 / a 2  +    2 / b 2  = 1 :

 

                                                                              k 2  = m 2 a 2 + b 2 .

 

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