HYPERBOLA

Definition: Locus of point which moves in a plane such that its distance from a fixed point is in a constant ratio to its corresponding distance from a fixed line, where constant ratio
e > 1 is called hyperbola.
 Standard equation of Hyperbola is    
     1) Centre (0, 0)
     2) Vertices A(a, 0), A'(-a, 0)
    3) Foci (ae, 0), (-ae, 0)    
    4) Directrices x = ± 
    5) Transverse axis is y = 0 (X − axis)
    6) Conjugate axis is x = 0 (Y − axis)
   7) If P be a point on the hyperbola drawn PN perpendicular to the transverse axis of hyperbola and produced to meet the curve again at P' then PP' is called a double ordinate.

Latus rectum: If double ordinate passing through focus then that is called Latus rectum.
Length of Latus rectum =  
End point of Latus rectum are  
 

Focal chord: A chord of the hyperbola passing its focus is called a Focal chord.
Auxiliary circle: Let   be hyperbola then the circle whose extremities
of diameter are vertices of hyperbola is called Auxiliary circle.
Equation of auxiliary circle of hyperbola is x² + y² = a²

 

PARAMETRIC EQUATION OF HYPERBOLA
Let P be a point on the hyperbola and PN is ordinate. Draw the tangent from N to
the auxiliary circle. Let Q be the point of contact, the line OQ makes an angle 'θ'
with transverse axis in +ve direction then ON = a sec
  
y₁² = b² tan²     y₁ = b tan ;   P (x₁, y₁) = (a sec , b tan

)
x = a sec θ, y = b tan are called parametric equations of hyperbola.
* Position of points Q on auxiliary circle and the corresponding point which describes   
the hyperbola 0 ≤ < 2π.
 

* Let P be any point on the hyperbola and S, S' are foci. SP, S'P are called focal distances.  
P (x₁, y₁) be any point on the hyperbola and S(ae, 0), S'(−ae, 0) are foci.
Then SP = ex₁ - a, S'P = ex₁ + a
          => S'P -  SP = 2a
* Difference of focal distances of any point on the hyperbola is equal to length transverse axis 2a.
*   is called conjugate hyperbola of given hyperbola 

Notations: 

 
                     
 * P(x₁, y₁), Q (x₂, y₂) are two points on the hyperbola then equation of chord passing through P(x₁, y₁), Q (x₂, y₂) is S₁ + S₂ = S₁₂
 P(a sec α, b tan α), Q(a sec β, b tan β) equation of chord passing through PQ is 
      
If the above chord passing through (ae, 0)
Then tan 
If the above chord passing through (−ae, 0), then tan  
* The point P(x₁, y₁) lies outside, on, or inside the hyperbola  according as
S₁<=> 0
* If y = mx + c is tangent to the hyperbola then c² = a²m² − b²
* For any value of m, y = mx ±  is tangent to  point of contacts are  

 * Equation of normal at P(x₁, y₁) to the hyperbola  

 is   = a² + b²

* Equation of normal to the hyperbola   
       at (a sec , b tan ) is   = a² + b²
* Equations of the normals of slope m to the hyperbola   are 
     
   at the points 

* If the straight line lx + my + n = 0 is normal to the hyperbola , then 
      
* In general four normals can be drawn to a hyperbola from a point. If α, β, γ, δ are eccentric angles of these four conormal points, then α + β + γ + δ = (2n + 1)Π, n  z.
*  The combined equation of the pair of tangents drawn from a point (x₁, y₁) to the
hyperbola  is  S₁² = SS₁₁

 * The locus of point of intersection of perpendicular tangents to hyperbola is x² + y² = a² − b², this is real circle if a > b, imaginary if b > a. This circle is known as director circle.
* The equation of chord of contact of tangents drawn from a point P(x₁, y₁) to the hyperbola    = 0 is S₁ = 0  
    
* The equation of a chord of hyperbola  = 0 bisected at the point P(x₁, y₁) is 
      
* Locus of point of intersection of tangents drawn from extremity of the chords passing through the fixed point is a straight is called polar of that point and point is called pole of that line with respect to hyperbola.
* If P(x₁, y₁) is tangent then polar of P(x₁, y₁) with respect to hyperbola is S1 = 0 i.e., 
     
* Pole of the line lx + my + n = 0 with respect to hyperbola

  is 
* Polar of the focus is directrix.
* If the polar of P(x₁, y₁) passes through Q(x₂, y₂), then polar of Q passes through P and such points are called conjugate points.
* If the pole of the line l₁ x + m₁y + n₁ = 0 lies on l₂x + m₂y + n₂ = 0, then pole of l₂x + m₂y + n₂ = 0 lies on  l₁x + m₁y + n₁ = 0. Such lines are called conjugate lines.
* If P(x₁, y₁), Q(x₂, y₂) are conjugate points, then S₁₂ = 0.
* If l₁x + m₁y + n₁ = 0, l₂x + m₂y + n₂ = 0 are conjugate lines, then  a²l₁l₂ − b²m₁m₂ = n₁n₂.
* Locus of middle points of a system of parallel chords of a hyperbola is called diameter.
* Equation of diameter whose slope of parallel chords m is y =  
* Two diameters are said to be conjugate when each bisect all chords parallel to the other.
* If y = m₁x, y = m₂x are two conjugate diameters, then m₁m₂ =  
* If the diameter intersects the hyperbola at real points, then conjugate diameter intersects at imaginary points.
* If the diameter intersects the conjugate hyperbola at real points, then conjugate diameter intersects conjugate hyperbola at imaginary points.
* If the diameter intersects the hyperbola at real points, then conjugate diameter intersects conjugate hyperbola at real points.
* Foot of the perpendicular from foci to any tangent lies on auxiliary circle, and product of the perpendiculars is b² (square of semi conjugate axis).
* An asymptote to a hyperbola is a straight line, at a finite distance from the origin, to which the tangent to hyperbola tends as the point of contact goes to infinity.
                                                                  (or)
* If the line is tangent to the hyperbola at infinity then that is called asymptote.
* The equations of two asymptotes of hyperbola are y = ±  x      

 

H + H' = 2A
H − A = constant

*  Parallelogram formed by the tangents at the extremities of conjugate diameter of a hyperbola has its vertices lying on the asymptotes and is of constant area '4ab'.
* If the tangent at P to hyperbola intersects the asymptote at M, M' then midpoint of MM' is P and area of triangle OMM' is constant is 'ab'.
* P(a sec θ, b tan ), P'(−a sec , −b tan ) are extremities of diameter, then Q(a tan , b sec), Q'(-a tan , -b sec ) are extremities of conjugate diameter.
* If '' is angle between the asymptotes, then = 2 tan−1

*  If the angle between the asymptotes is 90^o, then that hyperbola is called rectangular hyperbola.

     
* Equation of hyperbola whose asymptotes are y = ± x is x² -  y² = a²
* Equation of hyperbola whose asymptotes are coordinate axes is xy = c²
* Parametric coordinates of rectangular hyperbola xy = c² is (ct,  ), t ≠ 0  R
* Eccentricity of rectangular hyperbola is .
* If a rectangular hyperbola with centre C, intersect a circle of radius r in four points P, Q, R, S then CP² + CQ² + CR² + CS² = 4r²
* If the vertices of a triangle lies on rectangular hyperbola, then its orthocentre also lies on that hyperbola.
* If a circle cuts a rectangular hyperbola xy = c² in A(ct₁, ), B(ct₂, ), C(ct₃, ) and D(ct₃, ) then t₁t₂t₃t₄ = 1 and the centre of mean position of the four points is equal to midpoint of centres of two curves.
* The equation of the tangent at (ct,  c/t) to the hyperbola xy = c² is    + yt = 2c
* Tangents at P(ct₁, ) and Q(ct₂, ) to the rectangular hyperbola xy = c² intersects at  

*  The equation of the normal at (x₁, y₁) to the hyperbola xy = c² is xx₁− yy₁ = x₁₂  − y₁₂ 
* Equation of the normal at (ct, c/t  to the hyperbola xy = c² is xt^₃ − yt − ct^₄ + c = 0
* If the normal at the point t₁, to the rectangular hyperbola xy = c² meets it again at the point t₂, then t₂ = 
* The normals at three points P, Q, R on a rectangular hyperbola intersect at a point
* T on the curve, then the centre of the hyperbola is the centroid of ∆ PQR.