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
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.
= 1 be the given hyperbola, then
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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