1 Standard Hyperbola :
| Hyperbola Imp. Terms |  |  |
| Center | (0,0) | (0,0) |
| Length of transverse axis | 2a | 2b |
| conjugate axis | 2b | 2a |
| Foci | (±ae ,0) | (0, ±be) |
| Equation of directories | x = ± a/e | y= ± b/e |
| Eccentricity |  |  |
| Length of L.R. Parametric | 2b2 /a | 2a2 /b |
| co-ordinates | (a src Æ, b tan Æ)
0 £ Æ < 2p | (b sec Æ, a tan Æ)
0 £ Æ < 2p |
| Focal radii | SP = ex1 - a
SP = ex1 + a | SP = ey1 - b
SP = ey1 + b |
| SP SP | 2a | 2b |
| Tangents at the vertices | x = - a, x = a | y= - b, y = b |
| Euation of the transverse axis | y =0 | x =0 |
| Equation of the conjugate axis | x=0 | y=0 |
2 Special form of hyperbola :
If the center of hyperbola is (h,k) and axes are parallel to the co-ordinate axes, then its equation is 
3 Parametric equation of hyperbola :
The equation x = a sec Æ and y = b tan Æ are known as the parametric equation of hyperbola (x2/a2) (y2/b2) =1
4 Position of a point and line w.r.t. a hyperbola :
(x2/a2) (y2/b2) =1 according as (x21 / a2) (y21 / b2) -1 is + ve zero ve
The line y = mx + c does not intersect, touches, intersect the hyperbola according as c2 < , = > a2m2 - b2.
5 Equation of tangent in different forms:
(a) point form: the equation of the tangent to the hyperbola (x2 / a2) (y2 / b2) = 1 at (x1, y1) is (xx1 /a2) yy1 / b2) =1
(b) Parametric form : The equation of tangent to the hyperbola
(x2 a2) (y2 / b2) =1 at (a sec Æ , b tan Æ) is (x/a) sec Æ - (y/b) tan Æ = 1.
(c) Slope form: the equation of tangent of slope m to the hyperbola
(x2 / a2) (y2 / b2) = 1 are y = mx ± Ö(a2m2 - b2) and the co-ordinates of points of contacts are
6 Equation of pair of tangents:Equation of pair of tangents from (x1, y1) to the hyperbola (x2 / a2) (y2 / b2) = 1 is given by SS1 = T2
7 Equations of normals in different forms :
(a) point : the equation of normal to the hyperbola
(x2 / a2) (y2 /b2) = 1 at (x1, y1) is (a2x / x1) + (b2y / y1) = a2 + b2 .
(b) Parametric form : the equation of normal at ( a sec q , b tan q) to the hyperbola
(x2 / a2) (y2 / b2) = 1 is ax cos q + by cot q = a2 + b2
(c) Slope form: the equation of the normal to the hyperbola (x2 / a2) (y2 / b2) = 1 in terms of the slope m of the normal is y = mx ± 
(d) Condition for normality : If y = mx + c is the normal of (x2 / a2) (y2 / b2) =1
Is condition of normality.
(e) Points of contact : co-ordinates of point of contact
8 The equation of director circle of hyperbola:
(x2 /a2) (y2 / b2 = 1 is x2 + y2 = a2 -b2.
9 Equation of chord of contact of the tangents :Equation of chord of contact of the tangents drawn form the external point (x1, y1) to the hyperbola is given by (xx1 / a2) (yy1 / b2) =1
10 The Equation of chord of the hyperbola :the equation of chord of the hyperbola (x2 / a2) (y2 b2) =1 whose mid point is (x1, y1) is T = S1.
11 Equation of chord joining the points:Equation of chord joining the points P(a sec Æ1, b tan Æ1) and Q( asec Æ2, b tan Æ2 ) is
12 Equation of polar of the point:Equation of polar of the point (x1, y1) w.r.t. the hyperbola is given by T =0. The pole of the line ex + my + n =0 w.r.t.
13 The equation of a diameter of the hyperbola:The equation of a diameter of the hyperbola (x2 / a2) (y2/ b2) =1 is y = (b2 / a2m)x.
14 The diameters:The diameters y = m1x and y = m2x are conjugate if m1m2 = (b2 / a2)
15 Asymptotes of a hyperbola :
* The equation of asymptotes of the hyperbola
(x2 / a2) (y2 / b2) =1 are y = ± (b/a) x
Asymptote to a curve touches the curve at infinity.
* The asymptote of a hyperbola passes though the center of the hyperbola .
* The combined equation of the asymptotes of the hyperbola (x2 / a2) (y2 / b2) = 1 is (x2 / a2) (y2 / b2) = 0
* The angle between the asymptotes of
(x2 / a2) (y2 / b2) = 1 is 2 tan-1 (y2 / b2) or 2 sec-1 e .
* A hyperbola and its cojugate hyperbola have the same asymptotes .
* The bisector of the angles between the asymptotes are the coordinate axes.
* Equation of hyperbola Equation of asymptotes = Equation of asymptotes Equation of conjugate hyperbola = constant .
16 Rectangular or Equilateral Hyperbola :
* A hyperbola for which a = b is said to be rectangular hyperbola, its equation is x2 - y2 = a2
* xy = c2 represents a rectangular hyperbola with asymptotes x = 0, y =0.
* Eccentricity of rectangular hyperbola is Ö2 and angle between asymptotes of rectangular hyperbola is 90°
* parametric equation of the hyperbola xy = c2 are
x = ct, y = (c/t) , where t is a parameter.
* Equation of chord joining t1, t2 on xy = c2 is x + y t1t2 = c(t1 + t2)
* Equation of tangent at (x1, y1) to xy = c2 is (x/x1) + (y/y1) = 2 Equation of tangents at t is x + yt2 = 2ct
* Equation of normal at (x1, y1) to xy = c2 is xx1 - yy1 = x21 - y21
* Equation of normal at t on xy = c2 is
Xt3 - yt ct4 + c = 0
(This result shows that four normal can be drawn form a point to the hyperbola xy = c2)
* If a triangle is inscribed in a rectangular hyperbola then its orthocenter lies on the hyperbola .
* Equation of cjord of the hyperbola xy = c2 whose middle point is given is T = S1
*Point of intersection of tangents at t1 & t2 to the
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