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- 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 Sâ€™ = (-ae, 0)
- Equations of the directrices are given by x = a/e and x = -a/e
- The coordinates of vertices are Aâ€™ = (-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 âˆš2 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 Î¸and y = b tan Î¸ together represent the hyperbola x
^{2}/a^{2}- y^{2}/b^{2}= 1. Here Î¸ 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} **Equations of tangents**

1. 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

2. Equation of tangent to hyperbola x^{2}/a^{2}- y^{2}/b^{2 }= 1at the point (a sec Î¸, b tan Î¸) is (x sec Î¸)/a - (y tan Î¸)/b= 1

3. y = mx Â±[(a^{2}m^{2}- b^{2}] can also be taken as the tangent to the hyperbola x2/a2 - y2/b2 = 1**Equations of normal**

1. Equation of normal to the hyperbola x^{2}/a^{2}- y^{2}/b^{2}= 1 at the point (x1, y1) is a^{2}x/x_{1}+ b^{2}y/y_{1 }= a^{2}- b^{2}= a^{2}e^{2}

2. Equation of normal at the point P(a sec Î¸, b tan Î¸) on the hyperbola x^{2}/a^{2}- y^{2}/b^{2}= 1 is ax/sec Î¸ + by/ tan Î¸ = 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 - Ã¼ In a hyperbola b
^{2}= a^{2}(e^{2}â€“ 1). In the case of rectangular hyperbola (i.e., when b = a) result become a^{2}= a^{2}(e^{2}â€“ 1) or e^{2}= 2 or e =âˆš2 - Ã¼ In parametric form, its coordinates are x= ct, y = c/t, t âˆˆ R ~ {0}
- Ã¼ 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}

1. Equation of tangent at P (x_{1}, y_{1}) is x/x_{1 }+ y/y_{1}= 2

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

3. 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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