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Some Important Facts and Definitions about Hyperbola

  1. Geometrical meaning of b
The hyperbola meets the y-axis, i.e. x = 0, where .
Thus, the hyperbola meets the y-axis in imaginary points, that is, the hyperbola does not meet the y-axis in real points. However, if we take two points B1 and B2 on y-axis on opposite sides of the origin such that OB1 = OB2 = b, then B1B2 is called the conjugate axis of the hyperbola.
  1. Definitions
Vertices
The points V1 and V2 are called the vertices of the hyperbola.

Transverse Axis
V1V2 is called the transverse axis of the hyperbola.

Conjugate Axis
B1 B2 is called the conjugate axis of the hyperbola.

Principal Axes
The transverse and conjugate axes together are called the principal axes of the hyperbola.

Centre
O is called the centre of the hyperbola. It is the point of intersection of the transverse and conjugate axes. It bisects every chord of the hyperbola that passes through it.
  1. Symmetry
Since the equation of the hyperbola contains only even powers of x as well as of y, it is symmetric about both the axes.
  1. Intersection with the axes
The hyperbola meets the x-axis y = 0 whereTherefore, the hyperbola meets the x-axis in V1(-a, 0) and V2 (a, 0).The hyperbola meets the y-axis x = 0 whereTherefore, the hyperbola does not meet the y-axis in real points.
  1. The hyperbola has a second focus and a second directrix

Let the equation of the hyperbola be .Since it is symmetrical about the y-axis therefore, if we take the points F2 and M2 on the positive side of the origin such that OF1 = OF2 and OM1 = OM2, then the point F2 will also be a focus of the hyperbola and M2, the line through M2 will be corresponding directrix. Since OF1 = OF2 and F1, F2 lie on the opposite side of O therefore coordinates of the foci are (
ae, 0).Since OM1 = OM2 and M1 and M2 and on the opposite side of O, equation of two directrices are x = .
  1. If we take the transverse axis along the y-axis, focus as (0, ae) and directrix as , then the equation to the hyperbola is given by In this case, the other focus is (0, - ae) and the corresponding directrix is .





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