A conic section is said to be a hyperbola
if its eccentricity is greater than 1

###

Four standard forms of hyperbola:

Four standard forms of hyperbola:

Eccentricity of four standard form of hyperbola is same and given below.

###
__Content__ __I__ __II__ __III__ __IV__

1. Equation x

where b^{2}/a^{2}-y^{2}/b^{2}=1 x^{2}/a^{2}-y^{2}/b^{2}=-1 (x-α)^{2}/a^{2}-(y-β)^{2}/b^{2}=1 (x-α)^{2}/a^{2}-(y-β)^{2}/b^{2}=-1^{2}=a

^{2}(e

^{2}-1) where a

^{2}=b

^{2}(e

^{2}-1) where b

^{2}=a

^{2}(e

^{2}-1) where a

^{2}=b

^{2}(e

^{2}-1)

2. Centre(C) (0,0) (0,0) (α,β) (α,β)

3. Vertices A,A’=(±a,0 ) B,B’=(0,±b) (α±a, β) (α,b± β)

4 . Foci(S,S’) (±ae,0) (0,±be) (α±ae,β) (α,β±be)

5. Z,Z’ (±a/e,0) (0,±b/e) (α±a/e,β) (α,β±b/e)

6. Ends of
LR (±ae, ±b

^{2}/a) (±a^{2}/b,±be) (α±ae, β±b^{2}/a) (α±a^{2}/b, β±be)
7. Eqn’s. of
Trans- y=0 x=0 y=β x=α

verse axis

verse axis

8. Eqn’s. of Conjugate x=0 y=0 x=α y=β

axis

axis

9. Eqn’s.Latas rectas x=±ae y=±be x=α±be y= β±be

10. Eqn’s of
Directrices x=±a/e y=±b/e x= α±a/e
y= β±b/e

11. Length of
Transverse 2a 2b 2a 2b

axis

axis

12. Length of Conjugate 2b 2a 2b 2a

axis

axis

13. Length of Latus Rectum 2 b

^{2}/a 2 a^{2}/b 2 b^{2}/a 2 a^{2}/b
14. Differences
of |S’P-SP|=2a |S’P-SP|=2b |S’P-SP|=2a |S’P-SP|=2b

focal distance(focal radii)

of a point P on the Hyperbola

focal distance(focal radii)

of a point P on the Hyperbola

16. Distance b/w the vertices AA’=2a BB’=2b AA’=2a BB’=2b

17. Distance
b/w directrices ZZ’=2a/e ZZ’=2b/e ZZ’=2a/e ZZ’=2b/e

...........................................................................................................................................................................................

II. The hyperbola x

^{2}/a

^{2}-y

^{2}/b

^{2}=1 and x

^{2}/a

^{2}-y

^{2}/b

^{2}=-1 are called conjugate hyperbolas to each other .

III. If e

_{1},e

_{2 }are the two eccentricities of two conjugate hyperbolas then e

_{1}

^{2}+ e

_{2}

^{2}= e

_{1}

^{2}e

_{2}

^{2}

IV. A hyperbola is said to be a rectangular hyperbola if the length of its transverse axis is equal to length of its conjugate axis

V. x

^{2}-y

^{2}=a

^{2},xy=c

^{2}represents a rectangular hyperbola

VI. The eccentricity of a rectangular hyperbola is

VII. We use the following notation in this chapter

S= x

^{2}/a^{2}-y^{2}/b^{2}-1, S’= xx_{1}/a^{2}-yy_{1}/b^{2}-1, S_{11=}x_{1}^{2}/a^{2}-y_{1}^{2}/b^{2}-1, S_{12}=xx_{1}/ a^{2}-yy_{1}/ b^{2}-1IX.A point is said (x

_{1,}y

_{1}) is said to be 0:-an external point to the hyperbola S=0 if S

_{11}<0.

ii) An internal point to the hyperbola S=0
if S

_{11}>0
iii) Lies on the hyperbola S=0 if S

_{11}=0X. Two tangents can be drawn to a hyperbola from an external point.

XI. The equation of the tangent to a hyperbola S=0 at P(x

_{1,}y

_{1}) is S

_{1}=0

XII. The equation of the normal to the hyperbola x

^{2}/a

^{2}-y

^{2}/b

^{2}=1 at P(x

_{1},y

_{1}) is a

^{2}x/x

_{1}+ b

^{2}y/b

_{1}=a

^{2}+b

^{2}

XIV. The condition that the line y=mx+c may be a tangent to the hyperbola x

^{2}/a

^{2}-y

^{2}/b

^{2}=1 is c

^{2}=a

^{2}m

^{2}-b

^{2}and the point of contact is (-a

^{2}m/c , -b

^{2}/c)

XV. The condition that the line lx+my+n=0 may be a tangent to the hyperbola x

^{2}/a

^{2}-y

^{2}/b

^{2}=1 is (-a

^{2}l/n,-b

^{2}m/n).

^{ }

XVI. The equation of a tangent to the hyperbola x

^{2}/a

^{2}-y

^{2}/b

^{2}=1 may be taken as y=mx±

XVII. If m

_{1},m

_{2}are the slopes of the tangents through P to the hyperbola x

^{2}/a

^{2}-y

^{2}/b

^{2}=1 then m

_{1}+m

_{2}=2x

_{1}y

_{1}/x

_{1}

^{2}-a

^{2 }; m

_{1}m

_{2}=y

_{1}

^{2}+ b

^{2}/ x

_{1}

^{2}-a

^{2 }

XVII. if

_{1},y

_{1}) to the hyperbola S= x

^{2}/a

^{2}-y

^{2}/b

^{2}-1=0 then

XIX. The equation to the director circle x

^{2}/a

^{2}-y

^{2}/b

^{2}=1 is x

^{2}+y

^{2}=a

^{2}-b

^{2}

XX. The equation to the auxiliary circle of x

^{2}/a

^{2}-y

^{2}/b

^{2}=1 is x

^{2}+y

^{2}=a

^{2}

XXI. The equation to the chord of contact of P(x

_{1},y

_{1}) with respect to the hyperbola S=0 is S

_{1}=0

XXII. The qquation to the chord of the hyperbola S=0 having P(x

_{1},y

_{1}) as its midpoint is S

_{1}=S

_{12}

XXIV. The midpoint of the chord lx+my+n=0 of the hyperbola x

^{2}/a

^{2}-y

^{2}/b

^{2}=1 is (-a

^{2}ln/ a

^{2}l

^{2}-b

^{2 }m

^{2},b

^{2}mn/ a

^{2}l

^{2}-b

^{2 }m

^{2 })

XXV. The equation to the pair of tangents to the hyperbola S=0 from P(x

_{1},y

_{1}) is S

_{1}

^{2}=SS

_{11}

XXVI. The equation x=asec

^{2}/a-y

^{2}/b

^{2}=1 and the point (asec

XXVII. If P(x

_{1},y

_{1}) =(asec

^{2}/a

^{2}-y

^{2}/b

^{2}=1 and its foci are S,S’ then

SP=|ex
and S’P=|ex

_{1}-a|=|asec_{1}+a|=|asecXXVII. The equation of the chord joining two points a and b on the hyperbola x

^{2}/a

^{2}-y

^{2}/b

^{2}=1 is x/a cos(α-β/2)-y/bsin(α+ β/2)=Cos (α+ β/2)

XXX. If a and b are the ends of a focal chord of a hyperbola S=0 then e cos(α- β/2)= Cos (α+ β/2)

XXXI. The equation of tangent at p(

XXXII. The equation of normal at p(

XXXIII. The condition that the line lx+my+n=0 to be a normal to the hyperbola x

^{2}/a

^{2}-y

^{2}/b

^{2}=1 is a

^{2}/ l

^{2}-b

^{2}/m

^{2}=( a

^{2}+ b

^{2})

^{2}/n

^{2}

XXXIV. Atmost four normals can be drawn from a point to a hyperbola

XXXV. The parametric equation of xy=c

^{2}are x=ct;y=c/t

###
__Properties of Asymptotes__:

i). The equation of the asymptotes of the hyperbola S=0 are x/a±y/b=0 (or) y=±b/ax

ii). The equation to the pair of asymptotes of x

iii). Equation of hyperbola and equation of its pair of asymptotes are differ in their constant terms only.

iv). Asymptotes of the hyperbola passes through the centre of the hyperbola and they are equally inclined to the axes of the hyperbola

v). The angle between the asymptotes of the hyperbola S=0 is 2
or 2

vi). The angle between the asymptotes of a rectangular hyperbola is

vii). The equation of a rectangular hyperbola whose asymptotes are the coordinate axes is xy=c

viii). The product of perpendiculars from any point on hyperbola S=0 to its asymptotes is a

ix). Asymptotes of a hyperbola and its conjugate hyperbola are same.

x). If H,C and A are the equation of a hyperbola and its pair of asymptotes respectively then H+C=2A

xi). Equation of pair of asymptotes of hyperbola ax
=0

ii). The equation to the pair of asymptotes of x

^{2}/a^{2}-y^{2}/b^{2}=1 is x^{2}/a^{2}-y^{2}/b^{2}=0iii). Equation of hyperbola and equation of its pair of asymptotes are differ in their constant terms only.

iv). Asymptotes of the hyperbola passes through the centre of the hyperbola and they are equally inclined to the axes of the hyperbola

v). The angle between the asymptotes of the hyperbola S=0 is 2

vi). The angle between the asymptotes of a rectangular hyperbola is

vii). The equation of a rectangular hyperbola whose asymptotes are the coordinate axes is xy=c

^{2}viii). The product of perpendiculars from any point on hyperbola S=0 to its asymptotes is a

^{2}b^{2}/a^{2}+b^{2}ix). Asymptotes of a hyperbola and its conjugate hyperbola are same.

x). If H,C and A are the equation of a hyperbola and its pair of asymptotes respectively then H+C=2A

xi). Equation of pair of asymptotes of hyperbola ax

^{2}+2hxy+by^{2}+2gx+2fy+c=0 is ax^{2}+2hxy+by^{2}+2gx+2fy+c-