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

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

Equation of 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

Equation of Hyperbola


  In this case,

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

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

Equation of Hyperbola Assignments

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

(iv)    Centre O (0, 0).

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

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(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 =

Equation of Hyperbola

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

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(5)     Equation of tangent in slope form is

y = mx ±

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

if c = ±

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  and point of contact is


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

Equation of Hyperbola

    (7)     Equation of chord of contact at point


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

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(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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