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Revision Notes on Hyperbola

  • The standard equation of hyperbola with reference to its principal axis along the coordinate axis is given by x2/a2 - y2/b2 = 1, where b2 = a2 (e2 -1).

  • The foci of the hyperbola are S(ae, 0) and S’ = (-ae, 0).

  • Equations of the directrices are given by x = a/e and x = -a/e.

  • The coordinates of vertices are A’ = (-a, 0) and A = (a, 0).

  • The length of latus rectum is 2b2/a = 2a(e2 - 1).

  • The length of the transverse axis of the hyperbola is 2a.

  • The difference of the focal distances of any point on the hyperbola is constant and is equal to transverse axis.

  • x2/a2 - y2/b2 = 1 and -x2/a2 + y2/b2 = 1 are conjugate hyperbola of each other.

  • If e1 and eare the eccentricities of the hyperbola and its conjugate then e1-2 + e2-2 = 1

  • The foci of a hyperbola and its conjugate are concyclic and form the vertices of a square.

  • The length of the transverse axis of a hyperbola is 2a and the transverse axis and conjugate axis together constitute the principal axis of the hyperbola.

  • Whether two hyperbolas are similar or not is decided on the basis of their eccentricity. The hyperbolas with same eccentricity are same.

Hyperbola

  • The equation of the auxiliary circle of the hyperbola is given by

x2+ y2 = a2

  • In parametric form, the equations x = a sec θ and y = b tan θ together represent the hyperbola x2/a2 - y2/b2 = 1. Here θ is a parameter.

  • The point P(x1, y1) lies within, on or outside the ellipse according as

x12/a2 - y12/b2 = 1 is positive, zero or negative.

  • The line y = mx + c is a secant, a tangent or passes outside the hyperbola  x2/a2 - y2/b2 = 1 according as whether c2 is > = or < a2m2 - b2

  • ?Equation of tangent:

  1. Equation of tangent to hyperbola x2/a2 - y2/b2 = 1at the point (x1, y1) is xx1/a2 - yy1/b2 = 1

  2. Equation of tangent to hyperbola x2/a2 - y2/b2 = 1 at the point (a sec θ, b tan θ) is (x sec θ)/a - (y tan θ)/b = 1

  3. y = mx ± [(a2m- b2] can also be taken as the tangent to the hyperbola x2/a2 - y2/b2 = 1

  • Equation of normal:

  1. Equation of normal to the hyperbola x2/a2 - y2/b2 = 1 at the point (x1, y1) is ax/x+ by/y1 = a+ b2 = a2e2

  2. Equation of normal at the point P(a sec θ, b tan θ) on the hyperbola x2/a2 - y2/b2 = 1 is ax/sec θ + by/ tan θ = a+ b2 = a2e2

  • ?The combined equation of pair of tangents drawn from a point P(x1, y1) lying outside the hyperbola x2/a2 – y2/b2 = 1 is SS1 = T2, where S = x2/a2 – y2/b2 - 1, x12/a2 – y12/b2 - 1 and T = xx1/a2 – yy1/b2 - 1

  • The tangent and normal at any point of a hyperbola bisect the angle between the focal radii.

  • The portion of the tangent between the point of contact and the directrix subtends a right angle at the corresponding focus.

  • The equation of director circle is x2 + y2 = a2 - b2  

  • The equations of the asymptotes of the hyperbola are
    x/a + y/b = 0 and x/a - y/b = 0.

    This can be combined as x2/a2 - y2/b2 = 0

  • The asymptotes of the hyperbola and its conjugate are same.

  • Comparison between hyperbola and its conjugate hyperbola:?

Basic Elements

Hyperbola

Conjugate Hyperbola

 

x2/a2 - y2/b2 = 1

x2/a2 - y2/b2 = -1

Centre

(0,0)

(0,0)

Length of transverse axis

2a

2b

Length of conjugate axis

2b

2a

eccentricity

b2= a2(e2-1)

a2= b2(e2-1)

Foci

(± ae,0)

(0, ± be)

Equation of directrix

x = ± a/e

y = ± b/e

Length of latus rectum

2b2/a

2a2/b

Difference of focal distances

2a

2b

Equation of transverse axis

y = 0

x = 0

equation of conjugates

x = 0

y = 0

Tangent at vertices

x = ± a

y = ± b

     
  • The asymptotes pass through the center of the hyperbola and the bisectors of the angles between the asymptotes are the axis of a hyperbola

  • If the lengths of transverse and conjugate axis are equal, then such a hyperbola is termed to be a rectangular hyperbola.

  • The eccentricity of rectangular hyperbola is √2 and the length of its latus rectum is equal to its transverse or conjugate axis. 

  • The equation xy = c2 represents a rectangular hyperbola

  • In a hyperbola b2 = a2 (e2 – 1). In the case of rectangular hyperbola (i.e., when b = a) result become a2 = a2(e2 – 1) or e2 = 2 or e =√2

  • In parametric form, its coordinates are x = ct, y = c/t, t ∈ R ~ {0}

  • Equation of a chord joining the points P(t1) and Q(t2) is x + t1t2y = c(t1 + t2) with slope m = -1/ t1t2

  • Equation of tangent at P (x1, y1) is x/x1 + y/y1 = 2

  • Equation of tangent at P(t) is x/t + ty = 2c

  • Equation of normal is y - c/t = t2(x – ct)

  • Chord whose middle point is given to be (h, k) is kx + hy = 2hk

  • The equation of the director circle of the hyperbola is given by x2 + y2 = a- b2.

  • If b< a2, this circle is real.

  • If b2 = a2 the radius of the circle is zero and it reduces to a point circle at the origin. In this case, centre is the only point from which the tangents at right angles can be drawn to the curve.

  • If b2 > a2 the radius of the circle is imaginary, so that there is no such circle and so no tangents at right angle can be drawn to the curve. 

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