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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 x^{2}/a^{2} - y^{2}/b^{2} = 1, where b^{2} = a^{2} (e^{2} -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 2b^{2}/a = 2a(e^{2} - 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.
x^{2}/a^{2} - y^{2}/b^{2} = 1 and -x^{2}/a^{2} + y^{2}/b^{2} = 1 are conjugate hyperbola of each other.
If e_{1} and e_{2}are the eccentricities of the hyperbola and its conjugate then e_{1}^{-2}+ e_{2}^{-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.
The eccentricity of rectangular hyperbola is √2 and the length of its latus rectum is equal to its transverse or conjugate axis.
The equation of the auxiliary circle of the hyperbola is given by x^{2}+ y^{2} = a^{2}
In parametric form, the equations x = a sec θand y = b tan θ together represent the hyperbola x^{2}/a^{2} - y^{2}/b^{2} = 1. Here θ is a parameter.
The point P(x_{1}, y_{1}) lies within, on or outside the ellipse according as x_{1}^{2}/a^{2} - y_{1}^{2}/b^{2} = 1 is positive, zero or negative.
The line y = mx + c is a secant, a tangent or passes outside the hyperbola x^{2}/a^{2} - y^{2}/b^{2} = 1 according as whether c^{2} is > = or < a^{2}m^{2} - b^{2}
Equation of tangent to hyperbola x^{2}/a^{2} - y^{2}/b^{2} = 1at the point (x_{1}, y_{1}) is xx_{1}/a^{2} - yy_{1}/b^{2} = 1
Equation of tangent to hyperbola x^{2}/a^{2} - y^{2}/b^{2} = 1at the point (a sec θ, b tan θ) is (x sec θ)/a - (y tan θ)/b= 1
y = mx ±[(a^{2}m^{2}- b^{2}] can also be taken as the tangent to the hyperbola x^{2}/a^{2} - y^{2}/b^{2} = 1
Equation of normal to the hyperbola x^{2}/a^{2} - y^{2}/b^{2} = 1 at the point (x_{1}, y_{1}) is a^{2 }x/x_{1}+ b^{2 }y/y_{1} = a^{2 }- b^{2} = a^{2}e^{2}
Equation of normal at the point P(a sec θ, b tan θ) on the hyperbola x^{2}/a^{2} - y^{2}/b^{2} = 1 is ax/sec θ + by/ tan θ = a^{2}+b^{2} = a^{2}e^{2}
The equation of director circle is x^{2} + y^{2} = a^{2} - b^{2}
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 x^{2}/a^{2} - y^{2}/b^{2} = 0
The asymptotes of the hyperbola and its conjugate are same
The asymptotes pass through the center of the hyperbola and the bisectors of the angles between the asymptotes are the axis of a hyperbola
The equation xy = c^{2} represents a rectangular hyperbola
ü In a hyperbola b^{2} = a^{2} (e^{2} – 1). In the case of rectangular hyperbola (i.e., when b = a) result become a^{2} = a^{2}(e^{2} – 1) or e^{2} = 2 or e =√2
ü In parametric form, its coordinates are x= ct, y = c/t, t ∈ R ~ {0}
ü Equation of a chord joining the points P(t_{1}) and Q(t_{2}) is
x + t_{1}t_{2}y = c(t_{1} + t_{2}) with slope m = -1/ t_{1}t_{2}
Equation of tangent at P (x_{1}, y_{1}) is x/x_{1} + y/y_{1} = 2
Equation of tangent at P(t) is x/t + ty = 2c
Equation of normal is y-c/t = t^{2}(x-ct)
Chord whose middle point is given to be (h, k) is kx + hy = 2hk
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