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Hyperbola. Focuses. Equation of hyperbola. Focal length.

Real and imaginary axes of hyperbola. Eccentricity.

Asymptotes of hyperbola. Equation of tangent line to hyperbola.

Tangency condition of straight line and hyperbola.


A hyperbola ( Fig.1 ) is called a locus of points, a modulus of difference of distances from which to the two given points  F1  and  F2 , called  focuses of hyperbola, is a constant value.


An equation of hyperbola ( Fig.1 ) is :


Here the origin of coordinates is a center of symmetry of hyperbola, and the coordinate axes are its axes of symmetry.

The segment  F1F2 = 2 ,  where   is called  a focal length. The segment  AB = 2 a  is called  a real axis of hyperbola, the segment  CD = 2 b is called an imaginary axis of hyperbola. The number  e = c / ae > 1 is called an eccentricity of hyperbola. The straight lines   y = ( b / a ) x  are called asymptotes of hyperbola.

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




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


                                                                                 k 2  = m 2 a 2 b 2 .



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