Monday, 13 March 2017

Hyperbola synopsis points




Definition:

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


Four standard forms of hyperbola:

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

Content           I                    II                        III                         IV


1.  Equation            x2/a2-y2/b2=1            x2/a2-y2/b2=-1                 (x-α)2/a2-(y-β)2/b2=1       (x-α)2/a2-(y-β)2/b2=-1   
                                 where b2=a2(e2-1)      where a2=b2(e2-1)          where b2=a2(e2-1)            where a2=b2(e2-1)      



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, ±b2/a)                   (±a2/b,±be)                     (α±ae, β±b2/a)              (α±a2/b, β±be)


7.  Eqn’s. of Trans-           y=0                                     x=0                            y=β                                    x=α
    verse axis


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


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



13. Length of Latus Rectum   2 b2/a                      2 a2/b                        2 b2/a                               2 a2/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



15. Distance between the foci         SS’=2ae              SS’=2be                       SS’=2ae                        SS’=2be


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     x2/a2-y2/b2=1 and  x2/a2-y2/b2=-1 are called conjugate hyperbolas to each other .


III. If e1,e2  are the two eccentricities of two  conjugate hyperbolas then e12+ e22= e12 e22


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. x2-y2=a2,xy=c2  represents a rectangular hyperbola


VI. The eccentricity of a rectangular hyperbola is


VII. We use the following notation in this chapter
        S= x2/a2-y2/b2-1,  S’= xx1/a2-yy1/b2-1, S11=x12/a2-y12/b2-1,  S12=xx1/ a2-yy1/ b2-1


IX.A point is said  (x1,y1) is said to be 0:-an external point to the hyperbola S=0   if  S11<0.
    ii) An internal point to the hyperbola S=0 if S11>0
    iii) Lies on the hyperbola S=0 if S11=0


X. 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(x1,y1) is S1=0


XII. The equation of the normal to the hyperbola  x2/a2-y2/b2=1    at   P(x1,y1) is a2x/x1 + b2y/b1=a2+b2


XIV. The condition that the line y=mx+c may be a tangent to the hyperbola x2/a2-y2/b2=1 is                        c2=a2m2-b2 and the point of contact is (-a2m/c , -b2/c)


XV. The condition that the line lx+my+n=0 may be a tangent to the hyperbola x2/a2-y2/b2=1 is (-a2l/n,-b2m/n).


XVI. The equation of a tangent to the hyperbola x2/a2-y2/b2=1 may be taken as y=mx±


XVII. If m1,m2 are the slopes of the tangents through P to the hyperbola x2/a2-y2/b2=1 then                          m1+m2=2x1y1/x12-a; m1m2=y12+ b2/ x12-a


XVII. if  is the angle between the tangents  drawn from a point (x1,y1) to the hyperbola S= x2/a2-y2/b2-1=0 then  



XIX. The equation to the director circle x2/a2-y2/b2=1 is x2+y2=a2-b2


XX. The equation to the auxiliary circle of  x2/a2-y2/b2=1 is x2+y2=a2


XXI. The equation to the chord of contact  of P(x1,y1) with respect to the hyperbola S=0 is S1=0


XXII. The qquation to the chord of the hyperbola S=0 having P(x1,y1)  as its midpoint is S1=S12


XXIV. The midpoint of the chord lx+my+n=0 of the hyperbola  x2/a2-y2/b2=1 is (-a2ln/ a2l2-b2 m2,b2mn/ a2l2-b2 m2  )


 XXV. The equation to the pair of tangents to the hyperbola S=0 from    P(x1,y1)   is S12=SS11


XXVI.  The equation x=asec   ; y=b tan  are called parametric equations of the hyperbola x2/a-y2/b2=1                           and the point (asec  b tan ) is called parametric point it is denoted by p.


XXVII. If  P(x1,y1) =(asec  b tan ) is a point on hyperbola x2/a2-y2/b2=1 and its foci are S,S’ then
             SP=|ex1-a|=|asec and S’P=|ex1+a|=|asec


XXVII. The equation of the chord joining two points a and b on the hyperbola x2/a2-y2/b2=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( ) on the hyperbola S=0 is x/asec  - y/btan =1


XXXII.  The equation of normal at  p( ) on the hyperbola S=0 is ax/sec +by /btan =1


XXXIII. The condition that the line lx+my+n=0 to be a normal to the hyperbola x2/a2-y2/b2=1  is  a2/                l2-b2/m2=( a2+ b2)2/n2



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


XXXV. The parametric equation of xy=c2 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 x2/a2-y2/b2=1   is x2/a2-y2/b2=0

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=c2

viii). The product of perpendiculars from any point on hyperbola S=0 to its asymptotes is a2b2/a2+b2

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 ax2+2hxy+by2+2gx+2fy+c=0 is                                        ax2+2hxy+by2+2gx+2fy+c- =0