HYPERBOLA A Conic is said to be a hyperbola if its eccentricity is greater than one.  If S ax22 hxy  by 2 2 gx  2 fy c 0 represents a hyperbola then h2  ab  0 and   0 2 2  If S ax2 hxy  by 2 gx  2 fy c 0 represents a rectangular hyperbola then h2  ab  0 ,   0 and a b  0 x2 y 2 Hyperbola: Equation of the hyperbola in standard form is  1. a2 b2

 Hyperbola is symmetric about both the axes.  Hyperbola does not passing through the origin and does not meet the Y - axis.  Hyperbola curve does not exist between the lines x  a and x  a . x2 y 2 xx yy x2 y 2 x x y y Notation: S   1 , S1  1  1, S1  1  1, S1 2  1 2  1. a2 b 2 1 a2 b 2 11 a2 b 2 12 a2 b 2 Rectangular hyperbola (or) Equilateral hyperbola : In a hyperbola if the length of the transverse axis (2a) is equal to the length of the conjugate axis (2b) , then the hyperbola is called a rectangular hyperbola or an equilateral hyperbola.  The eccentricity of a rectangular hyperbola is 2 . Conjugate hyperbola : The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called the conjugate hyperbola of the given hyperbola.

2 2 1 x y  The equation of the hyperbola conjugate to S  0 is S   1 0 . a2 b 2

 If e1 and e2 be the eccentricities of the hyperbola and its conjugate hyperbola then 1 1 2  2 1. e1 e2

x2 y 2  Parametric Equations: The parametric equations of a hyperbola  1 are a2 b 2 x asec and y b tan where is parameter ( 0  2  ).A point on the hyperbola is represented by ( asec , b tan  ) ¨ Auxiliary : The circle described on the transverse axis of a hyperbola as diam- eter is called the auxiliary circle of the hyperbola  The locus of the feet of the from foci on any of the hyperbola is a circle concentric with the hyperbola. This circle is called Auxiliary circle of the hyperbola.  The equation of auxiliary circle of S=0 is x2+y2=a2  Director Circle: The locus of point of intersection of to a hyperbola is a circle concentric with hyperbola. This circle is called director circle of the hyperbola  The equation to the director circle of S  0 is x2 y 2  a 2  b 2

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SKVVMATH.COM HYPERBOLA x2 y 2 x2 y 2  1  1 Characteristics a2 b2 a2 b 2

a2 b 2 a2 b 2 Ecentricity e  e  a2 b2

Center (0 , 0) (0 , 0)

Vertices ( a,0) (0,b )

Foci ( ae,0) (0,be )

2 2 b 2 2 a Length of latusrectum a b

b 2 a 2 Ends of latusrectum ( ae,  ) ( , be ) a b

y be Equation of latusrectum x ae

Length of transversal axis 2a 2b

Length of conjugate axis 2b 2a

a b Equation of diretrices x   x   e e

Equation of transversal axis y  0 x  0

y  0 Equation of conjugate axis x  0

2 2 2 2 2 2 Equation of auxillary circle x y  a x y  b

2 2 2 2 2 2 2 2 Equation of director circle x y a b x y  b- a

The absolute difference of focal distances of any point on the hyperbola 2a 2b

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Position of a Point with respect to a hyperbola: Let P(x1 ,y 1 ) be a point and x2 y 2 S   1 0 be a hyperbola then a2 b 2

(i) p lies on the hyperbola iff S11  0 .

(ii) p lies inside (i.e. in the region not containing the centre) the hyperbola iff S11  0 .

(iii) p lies out side the hyperbola iff S11  0 .

Tangent to Hyperbola: The equation of the tangent at p(x1, y1) is S1  0 . x y  The equation of the tangent at " " is sec  tan 1.  a b  The condition for a straight line y  mx  c to be a tangent to the hyperbola S  0 is   a2m b2  2 2 2 2   c  a m  b and the point of contact is  ,  .  c c   The equation of the tangent to the hyperbola S  0 having slope m is b b 2 2 2 ,m, m  y  mx  a m  b a a

 Two tangents can be drawn to a hyperbola from an external point. x2 y 2  If m,m are the slopes of the tangents through P(x y ) to the hyperbola  1 1 2 1, 1 a2 b2 2x y y 2  b2 m  m  1 1 m m  1 then 1 2 2 2 , 1 2 2 2 . x1  a x1  a  The product of the perpendiculars from foci to any tangent to the hyperbola S  0 is b 2 .

Normal to Hyperbola: The equation of the normal to the hyperbola S  0 at (x1 , y 1 ) is 2 2 a x b y 2 2   a  b . x1 y1

ax by 2 2    The equation of the normal at  is   a  b .0, ,   sec tan 2   The condition that the line lx  my  n  0 may be normal to the hyperbola S  0 is a 2 b2 (a2  b2 )2   . l 2 m2 n2  The condition that the line y mx  c to be a normal to the hyperbola S  0 is m a2 b 2  c  a2 b 2

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x2 y 2 m a2 b 2   Equation of normal to the hyperbola  1 in slope form is y mx  a2 b 2 a2 b 2 Co-normal points: In general four normals can be drawn to a hyperbola from any point. If ,,, be the eccentric angles of these four co-normal points then      is an odd multiple of  i.e., Sum of the eccentric angles of four co-normal points is an odd multiple of  .  If ,,   are the eccentric angles of three co-normal points on the hyperbola x2 y 2  1 , the normals at which are concurrent, then a2 b 2 sin sin   sin   0 x2 y 2  The equation of the chord joining the two points  and  on the hyperbola  1 a2 b2 is x y      cos sin   cos a2 b  2 2 x2 y 2  If P x, y   a sec , b tan   is a point on  1 and SS,  are foci then. 1 1 a2 b 2

SP ex1 a aesec  a and S P ex1 a aesec  a x2 y 2  If Px , y  is a point on  1 0 and SS,  are foci then SP ey b; S' P  ey b 1 1 a2 b 2 1 1 Asymptotes :  The tangents at infinity to the hyperbola are called asymptotes of the hyperbola. x2 y 2  The pair of equation of the asymptotes of the hyperbola S  0 is   0 . a 2 b2  The equation of a hyperbola and its pair of equation of asymptotes always differ by a constant.  The asymptotes are always passing through the centre of the hyperbola.  Asymptotes are equally inclined to the axes of hyperbola.

1 b   The angle between the asymptotes of the hyperbola S  0 is 2sec1(e) or 2 tan   .  a   The product of the perpendiculars from anypoint on the hyperbola to its asymptotes is a2b2 . a 2  b2  Any straight line parallel to an asymptotes of a hyperbola intersects the hyperbola in only one point.  Asymptotes of a hyperbola and its conjugate hyperbola are same  If SS, ' and A are the equations of a hyperbola, its conjugate hyperbola and its pair of asymptotes respectively then SS'  2 A

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Rectangular Hyperbola x2 - y 2 = a 2 : The hyperbola in which the lengths of the transverse and conjugate axes are equal is called an equilateral or rectangular hyperbola ( a b ) (i) Eccentricity, e  2 (ii) The equation of the asymptotes are y x (iii) The angle between the asymtotes is 90 . Rectangular hyperbola xy = c2 : If we rotate the axes through an angle of 450 in clock wise direction x2 y 2  a 2 will

2 2 a  become xy c2 , c   . (i) centre 0,0 (ii) vertices c,  c (iii) Foci 2  2c ,  2 c (iv) e  2 (v) LLR 2 2 c (vi) axes y xy,  x (vii) Asymptotes x0, y  0 (viii) Angle between the asymptotes  900 2 (ix) Equation of tangent at  x1, y 1  is xy1 xy 1  2 c

2 2 (x) Equation of nomral at  x1, y 1  is xx1 yy 1  x 1  y 1 c (xi) Parametric equations xcty,  , t  R  0  t (xii) Equation of tangent at 't ' is x yt2 2 ct  0 3 4 (xiii) Equation of noraml at 't ' is xt yt ct 1 3 (xiv) Normal at t1 meets the curve again at t2 then t1 t 2 1 c  c   c  2 (xv) Pct1,,,,,  Qct 2  Rct 3  are points on xy c then the ortho centre of PQR t1  t 2   t 3  c  is S, cttt1 2 3  ttt1 2 3  Note : S also lies on xy c2 xiv) If a circle and a rectangular hyperbola xy c2 meet at four points

c  4 4 xyii,   ct i , , i  1,2,3,4   then tttt1 2 3 4  1 xxxx1 2 3 4  c, yyyy 1 2 3 4  c ti 

c c c   c Note: Pct1,,,,,,, Qct 2 Rct 3  Sct  4 are concyclic then the ortho centre of t1 t 2 t 3   t 4

c  PQR  ct4 ,   S t4   The hyperbola conjugate to xy c2 is xy c2  The equation of the hyperbola having x   and y   as asymptotes can be taken as  x y    k 2

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 Equation of the hyperbola with reference to two perpendicular lines ax1 by 1  c 1 0 and

2 2 axbyc  bxayc   1 1 1  1 1 2  2 2 2 2 bx ay  c  0 as axes of coordinates is a b  a  b  1 1 2 1 1  1 1  1 a2 b 2

 The equation to the chord of contact of Px 1, y 1  with respect to the hyperbola S  0 is

S1  0 .

 The equation of the chord of the hyperbola S  0 having Px 1, y 1  as its midpoint is

SS1 11

2  The equation of pair of tangents drawn from Px 1, y 1  to the hyperbola S  0 is SSS1 . 11  Reflection property of a hyperbola : If an incomming light ray passing through one focus and strike convex side of the hyperbola then it will get reflected toward other focus S

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