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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 m1m=  
* 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 90o, 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 xt3 − yt − ct4 + 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.

Posted Date : 19-02-2021

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