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Hyperbola

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THE HYPERBOLA

The equation of a hyperbola in the standard form is given by

 

x2/a2 - y2/b2 = 1

The relation between a and b is given by

 

b2 = a2(1-e2)

 
PROPERTIES OF A HYPERBOLA

  1. The points A and A’ are called the vertices of the hyperbola. The coordinates of A and A’ are given by (a,0) and (-a,0) respectively. The line AA’ is called the transverse axis of the hyperbola and its length is given by 2a.
  2. The line BB’ is called the conjugate axis of the hyperbola. The points B and B’ are the points on the y-axis such that CB + CB’ = b.
  3. The chord of the hyperbola, which is perpendicular to the transverse axis and which also passes through the focus of the hyperbola is called the latus rectum of the hyperbola. It is represented by the equation

    4.  

    X = ae

    Its length is given by 2b2/a, and the coordinates of the end point are given by

    L = (ae , b2/a) and L’ = (ae , -b2/a)

  4. Since the hyperbola is a symmetrical figure (as shown above ) there are two foci and two directrix. The coordinates of the second focus is given by (-ae,0), and the equation of the second directrix is given by x = -a/e.
  5. The focal distance of any point is defined as the distance between the focus and the point.

NOTE; the difference the focal distances of any point is always equal to the length of the transverse axis.

From the figure S’P - SP = 2a

6. The parametric coordinate of an hyperbola is given by ( a secq , b tanq ). This equation satisfies the equation of an hyperbola for all values of q .

NOTE: The equation of an hyperbola, for which the transverse axis lies along the y-axis is given by

 

x2/b2 - y2/a2 = 1

The shape of this type of a hyperbola is as shown in the figure.

   

TANGENTS AND NORMALS

TANGENTS

The tangent at any point, A = (x1,y1), on an hyperbola is given by the equation

 

x x1 /a2 - y y1/b2 = 1

The equation of the tangent in terms of slope is given by

 

y = mx + (a2m2 +b2)1/2

NORMALS

The normal at any point, A = (x1,y1), on an hyperbola is given by the equation

 

(x - x1 ) / x1/a2 = (y - y1)/ -y1/b2

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