Class 11 Maths Chapter 11 Conic Section Part 2 Hyperbola Download in PDF
Class 11 Maths Chapter 11 Conic Section Part 2 Hyperbola Download in PDF
Class 11 Maths Chapter 11 Conic Section Part 2 Hyperbola Download in PDF
The fixed point is called the focus and the fixed line is directrix and the ratio is the eccentricity.
The line through the foci of the hyperbola is called its transverse axis.
The line through the centre and perpendicular to the transverse axis of the hyperbola is called
its conjugate axis.
1. Centre O(0, 0)
2. Foci are S(ae,0),S1(-ae, 0)
3. Vertices A(a, 0), A1(-a, 0)
4. Directrices / : x = a/e, l : x = -a/e
5. Length of latusrectum LL1 = LL1 = 2b2/a
6. Length of transverse axis 2a.
7. Length of conjugate axis 2b.
8. Eccentricity or b2 = a2(e2 1)
9. Distance between foci =2ae
10. Distance between directrices = 2a/e
Conjugate Hyperbola
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The distance of a point on the hyperbola from the focus is called it focal distance. The
difference of the focal distance of any point on a, hyperbola is constant and is equal to the
length of transverse axis the hyperbola i.e.,
S1P SP = 2a
where, S and S1 are the foci and P is any point or P the hyperbola.
1 If the centre of the hyperbola is (h, k) and the directions of the axes are parallel to the
coordinate axes, then the equation of the hyperbola, whose transverse and conjugate axes are 2a
and 2b is
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2. If a point P(x, y) moves in the plane of two perpendicular straight lines a 1x + b1y + c1 = 0
and b1x a1y + c2 = 0 in such a way that
Then, the locus of P is hyperbola whose transverse axis lies along b1x a1y + c2 = 0 and
conjugate axis along the line a1x + b1y + c1 = 0. The length of transverse and conjugate axes are
2a and 2b, respectively.
Parametric Equations
x = a sec , y = b tan
or x = a cosh , y = b sinh
Equation of Chord
(i) Equations of chord joining two points P(a sec 1, b tan 1,) and Q(a sec 2, b tan 2) on the
hyperbola
(ii) Equations of chord of contact of tangents drawn from a point (x1, y1) to the
hyperbola
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(iii) The equation of the chord of the hyperbola bisected at point (x1, y1) is given
by
hyperbola
the hyperbola
(iv) The tangent at the points P(a sec 1 , b tan 1) and Q (a sec 2, b tan 2) intersect at the
point
(v) Two tangents drawn from P are real and distinct, coincident or imaginary according as the
roots of the equation m2(h2 a2) 2khm + k2 + b2 = 0. are real and distinct, coincident or
imaginary.
contacts
hyperbola
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(ii) Parametric Form The equation of the normal at (a sec , b tan ) to the
hyperbola
is ax cos + by cot = a2 + b2.
(iii) Slope Form The equations of the normal of slope m to the hyperbola are
given by
(v) Maximum four normals can be drawn from a point (x1, y1) to the hyperbola
Conormal Points
Points on the hyperbola, the normals at which passes through a given point are called conormal
points.
1. The sum of the eccentric angles of conormal points is an odd ion multiple of .
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4. If the normals at four points P(x1, y1), Q(x2, y2), R(x3 , y3) and S(X4, y4) on the
1. Two points are said to be conjugate points with respect to a hyperbola, if each lies on the
polar of the other.
hyperbola is
Asymptote
An asymptote to a curve is a straight line, at a finite distance from the origin, to which the
tangent to a curve tends as the point of contact goes to infinity.
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Director Circle
The locus of the point of intersection of the tangents to the hyperbolo , which are
perpendicular to each other, is called a director circle. The equation of director circle is x 2 +
y2 = a2 b2.
Rectangular Hyperbola
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6. Directrices x = + a/2
7. Latusrectum = 2a
8. Parametric form x = a sec , y = a tan
9. Equation of tangent, x sec y tan = a
1. Point Form The equation of tangent at (x1, y1) to the rectangular hyperbola is xy1 + yx1 =
2c2 or (x/x1 + y/y1) = 2.
2. Parametric Form The equation of tangent at (ct, c/t) to the hyperbola is( x/t + yt) = 2c.
3. Tangent at P(ct1, c/t1) and Q (ct2, c/t2) to the rectangular hyperbola intersect
a
4. The equation of the chord of contact of tangents drawn from a point (x 1, y1) to the
rectangular hyperbola is xy1 + yx1 = 2c2.
1. Point Form The equation of the normal at (x1, y1) to the rectangular hyperbola is xx1
yy1 = x12 y12.
2. Parametric Form The equation of the normal at ( ct, c/t)to the rectangular hyperbola xy
= c2 is xt3 yt ct4 + c = O.
3. The equation of the normal at( ct, c/t)is a fourth degree equation t in t. So, in general
four normals can be drawn from a point to the hyperbola xy = c2.
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1. The point (x1, y1) lies outside, on or inside the hyperbola according
as
2. The combined equation of the pairs of tangent drawn from a point P(x1, y1) lying outside
the hyperbola
3. The equation of the chord of the hyperbola xy = c2 whose mid-point is (x1, y1) is
xy1 + yx1 = 2x1y1
or t = S1
4. Equation of the chord joining t1, t2 on xy = t2 is
x + yt1t2 = c(t1 + t2)
5. Eccentricity of the rectangular hyperbola is 2 and the angle between asymptotes is 90.
6. If a triangle is inscribed in a rectangular hyperbola, then its orthocentre lies on the
hyperbola.
7. Any straight line parallel to an asymptotes of a hyperbola intersects the
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