# Some Important Facts and Definitions about Hyperbola

**Geometrical meaning of b**

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 B

_{1}and B

_{2}on y-axis on opposite sides of the origin such that OB

_{1}= OB

_{2}= b, then B

_{1}B

_{2}is called the conjugate axis of the hyperbola.

**Definitions**

**Vertices**

The points V

_{1}and V

_{2}are called the vertices of the hyperbola.

**Transverse Axis**

V

_{1}V

_{2}is called the transverse axis of the hyperbola.

**Conjugate Axis**

B

_{1}B

_{2}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.

**Symmetry**

**Intersection with the axes**

_{1}(-a, 0) and V

_{2}(a, 0).The hyperbola meets the y-axis x = 0 whereTherefore, the hyperbola does not meet the y-axis in real points.

**The hyperbola has a second focus and a second directrix**

_{2}and M

_{2}on the positive side of the origin such that OF

_{1}= OF

_{2}and OM

_{1}= OM

_{2}, then the point F

_{2}will also be a focus of the hyperbola and M

_{2, }the line through M

_{2}will be corresponding directrix. Since OF

_{1}= OF

_{2}and F

_{1}, F

_{2}lie on the opposite side of O therefore coordinates of the foci are (

ae, 0).Since OM

_{1}= OM

_{2}and M

_{1}and M

_{2}and on the opposite side of O, equation of two directrices are x = .

- 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 .