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Hyperbola

 

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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