Conic Sections - Hyperbola, Equation of Hyperbola, Conic Sections, Co-Ordinate Geometry

Hyperbola: When the ratio (defined in parabola and ellipse) is greater than 1, i.e., e > 1, then the conic is said to be hyperbola.

Since the equation of the hyperbola

  differs from that of the a b2, most of the results proved for the ellipse are true for the hyperbola, if we replace b² by - b² in their proofs. We therefore, give below the list of corresponding results applicable in case of hyperbola. 

(1)     Standard equation of hyperbola is

                   Where

  In this case,
       

(i)      Foci are S (ae, 0) and S' (- ae, 0).

(ii)     Equations of directrix ZM ,and Z'M'

 

(iii)    Transverse axis AA' = 2a, conjugate axis BB' = 2b.

(iv)    Centre O (0, 0).

(v)     Length of latus rectum LL' = L1L1'

(vi)    The difference of focal distances from any point P(x1, y1) on hyperbola remains constant and is equal to the length of transverse axis. i.e.,

S'P - SP = (ex1 + a) - (ex1 - a) = 2a.
 

(2)     The equation of rectangular hyperbola x² - y² = a² = b² i.e., in standard form of hyperbola put a = b.

Hence e =

for rectangular hyperbola.

(3)     Parametric equations for hyperbola are x = a sec θ and y = b tan θ

(4)     Equation of tangent at any point

(x1, y1) on hyperbola is

(5)     Equation of tangent in slope form is

y = mx ±

i.e., the line y = mx + c is a tangent to hyperbola,

if c = ±

  and point of contact is

 (6)     the equation of normal at any point (x1, y1) on hyperbola is

    (7)     Equation of chord of contact at point

(8)     Chord whose mid-point is (h, k) is

(9)     Chord joining 't1' and 't2' is x + yt1t2 - c(t1 + t2) = 0. 

(10)   tangent at t1 is x + yt12 - 2ct, = 0

 
                   where x1 = ct1,

 

(11)   The length of chord cut off by hyperbola from the line y = mx + c is

           

(12)   Equation of director circle is x² + y² = a² - b²
 

(13)   If Y = m1x and y = m₂x are the conjugate diameters of hyperbola, then

 
(14)   Conjugate Hyperbola: The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of the hyperbola is called the conjugate hyperbola of the given hyperbola.

Thus is

= 1 be the given hyperbola, then

  

 conjugate hyperbola.

 

If e and e' are the eccentricities of these two hyperbolas, then

   

(15)   Equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents a hyperbola, if Δ ¹ 0, h2 > ab and rectangular hyperbola if Δ ¹ 0, h2 > ab and a + b = 0.

 

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