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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

గమనిక : ప్రతిభ.ఈనాడు.నెట్లో వచ్చే ప్రకటనలు అనేక దేశాల నుండి, వ్యాపారస్తులు లేదా వ్యక్తుల నుండి వివిధ పద్ధతులలో సేకరించబడతాయి. ఆయా ప్రకటనకర్తల ఉత్పత్తులు లేదా సేవల గురించి ఈనాడు యాజమాన్యానికీ, ఉద్యోగస్తులకూ ఎటువంటి అవగాహనా ఉండదు. కొన్ని ప్రకటనలు పాఠకుల అభిరుచిననుసరించి కృత్రిమ మేధస్సు సాంకేతికతతో పంపబడతాయి. ఏ ప్రకటనని అయినా పాఠకులు తగినంత జాగ్రత్త వహించి, ఉత్పత్తులు లేదా సేవల గురించి తగిన విచారణ చేసి, తగిన జాగ్రత్తలు తీసుకొని కొనుగోలు చేయాలి. ఉత్పత్తులు / సేవలపై ఈనాడు యాజమాన్యానికి ఎటువంటి నియంత్రణ ఉండదు. కనుక ఉత్పత్తులు లేదా సేవల నాణ్యత లేదా లోపాల విషయంలో ఈనాడు యాజమాన్యం ఎటువంటి బాధ్యత వహించదు. ఈ విషయంలో ఎటువంటి ఉత్తర ప్రత్యుత్తరాలకీ తావు లేదు. ఫిర్యాదులు తీసుకోబడవు.

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