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Hyperbola

A hyperbola is a locus of a moving point such that the ratio of its distance from a fixed point(focus F)  to its distance from a fixed line(directrix l) is constant, i.e. it is a conic with eccentricity e>1.

The equation of hyperbola is

where b2=a2(e2-1)

The graph of the hyperbola is,

Terminology:

Centre: the point O

Vertices: the points A(a, 0) and A’(-a, 0)

Transverse axis: the segment joining the vertices; AA’above. Its length is 2a.

Conjugate axis: the line through the centre perpendicular to the transverse axis (the y-axis above). The hyperbola never intersects the conjugate axis. The length o f the conjugate axis is 2b.

Focus: the fixed point is called a focus F(ae,0) of the hyperbola.

Directrix: the fixed line is called the directrix of the hyperbola and its equation is x=a/e

Eccentricity: e=

Latus Rectum: the length of the latus rectum is . The end points of the latus rectum are (ae, ) and (ae, )

Properties:

Let P(x,y) be any point on the hyperbola.

From the above figure the focal distances PF=PF’=2a (the transverse axis)

Another definition of hyperbola:

A hyperbola is the locus of a moving point such that the difference of its distances from two fixed points is always constant. The two fixed points are called the foci of the hyperbola.

The other form of hyperbola is

For this hyperbola,

Centre: C (0,0)

Vertices: A (0, a), A’ (0, -a)

Foci: F (0, ae), F’ (0, -ae)

Equation of transverse axis is x=0

Equation of conjugate axis is y=0

End points of conjugate axis : ( b, 0) and (-b, 0)

Equation of latus rectum: y=± ae

Equation of directrices: y=

End points of the latus rectum: ( ± , ae),( ± , −ae)

 Cartesian form Parametric form Equation of chord at (x1, y1)and (x2,y2) y-y1= Chord joining the points  is Equation of tangent at (x1, y1) At Equation of normal at (x1, y1) At

Equations of two asymptotes to the hyperbola are

y= and y= -

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