7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 1/70
HYPERBOLA
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 2/70
Introduction
General Equation : ax2+2hxy+by2+2gx+2fy+c = 0denotes the hyperbola if h2 > ab and e > 1.
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 3/70
Standard Equation &
Basic TerminologyStandard equation of hyperbola is deduced using an
important property of hyperbola that the difference of a point moving on it, from two fixed points is
constant.i.e. |PF1 – PF
2| = 2a (2a < 2c i.e. > a)
. .
i.e.
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 4/70
Definitions
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 5/70
(i) Line containing the fixed point F1
and F2
(called
Foci) is called Transverse Axis (TA) of a FocalAxis and the distance between F1
and F2
is
called Focal Length.
(ii) The points of intersection (A1, A2) of the curvewith the transverse axis are called vertices of
the hyperbola.(iii) The length ‘2a’ between the vertices is called
the Len th of Transverse Axis.
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 6/70
(iv) The perpendicular bisector of transverse axis is
called the Conjugate Axis (CA). The pointB1(0,–b) and B
2(0,–b) which have special
significance, are known as the extermities of
conjugate axis and the length ‘2b’ is called theLength of conjugate axis. The point of
intersection of these two axes is called thecentre ‘O’ of the hyperbola. (Transverse axis
and con u ate axis to ether are called the
Principal Axis). Any chord passing throughcentre is called Diameter (PQ) and is bisected
by it.
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 7/70
(v) Any chord passing through focus is called is
Focal Chord and any chord perpendicular to theTransverse axis is called a Double Ordinate
(AB).
(vi) A particular double ordinate which passesthrough focus or a particular focal chord passing
through focus is called the Latus Rectum(L1L
2).
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 8/70
Eccentricity
Defines the curvature of the hyperbola and ismathematically spelled as :
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 9/70
Remember that
(i) a2e2 = a2 + b2
(ii) Coordinates of foci : (± ae, 0) and
have the same value of eccentricity.
(iv) Equation of hyperbola in terms of eccentricity
can be written as
x rem es o a us rec um
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 10/70
Conjugate Hyperbola
Corresponding to every hyperbola these exist ahyperbola such that, the conjugate axis and transverse
conjugate axis of other, such hyperbola are known
conjugate to each other.
* Hence for the hyperbola,
e con uga e yper o a s,
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 11/70
Q. If e1
and e2
are the eccentricities of a hyperbola
and its conjugate respectively, then prove thate1
–2 + e2
–2 = 1
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 12/70
Note :
The foci of a hyperbola and its conjugate are concylic
.
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 13/70
Focal Directrix Property
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 14/70
Rectangular HyperbolaIf a = b, hyperbola is said to be equilateral or
rectangular and has the equation x2 – y2 = a2.
ccen r c y or suc a yper o a sl(LR) 2a (e2 – 1) = 2a = l(Ta)
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 15/70
* Parametric coordinates
x = a secθ and y = b tanθ
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 16/70
Illustrations on Basic ParametersQ. On a level plain the crack of the rifle and the
thud of the ball striking the target are heard at
e same ns an ; prove a e ocus o ehearer is a hyperbola.
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 17/70
Q. Show that the locus of the centre of a circle
which touches externally two given circles is a
hyperbola.
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 18/70
Q. Given the base of a triangle and the ratio of the
tangents of half the base angles, prove that the
vertex moves on a hyperbola whose foci are the
extremities of the base.
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 19/70
Q. An ellipse and a hyperbola are confocal (have
the same focus) and the conjugate axis of
hyperbola is equal to the minor axis of the
ellipse, If e1
and e2
are the eccentricities of
ellipse and hyperbola then prove that
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 20/70
Q. Find the equation of hyperbola referred to its
principal axes as the coordinates axes
(a) If the distance of one of its vertices from the
foci are 3 and 1.
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 21/70
Q. (a) Whose centre is (1, 0) ; focus is (6, 0) and
transverse axis 6.
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 22/70
Q. (c) Whose centre is (3, 2), one focus is (5, 2)
and one vertex is (4, 2)
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 23/70
Q. (d) Whose centre is (-3, 2), one vertex is (-3, 4)
and eccentricity is 5/2.
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 24/70
Q. (e) Whose foci are (4, 2) and (8, 2) and
eccentricity is 2.
Q ( ) Fi d h di f h f i d h
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 25/70
Q. (e) Find the coordinates of the foci and the
centre of the hyperbola
Q Fi d h di f h f i d h
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 26/70
Q. Find the coordinates of the foci and the centre
of the hyperbola
Q Fi d thi f th h b l
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 27/70
Q. Find everything for the hyperbola
9x2 – 18x – 16y2 – 64y + 89 = 0
A i i Ci
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 28/70
Auxiliary Circle
A circle drawn with centre C & T.A. as a diameter iscalled the Auxiliary Circle of the hyperbola. Equation
o e aux ary c rc e s x y = a .
P i i f A P i ‘P’ A
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 29/70
Position of A Point ‘P’ w.r.t. A
Hyperbola
Li A d A H b l
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 30/70
Line And A Hyperbola
N t
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 31/70
Note
(i) For a given m, there can be two paralleltangents to the hyperbola
,a maximum of two tangents.
Di t Ci l
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 32/70
Director Circle
x2 + y2 = a2 – b2
(i) If l (TA) > l (CA) ; director circle is real with
.(ii) If l (TA) = l (CA) ; director circle is a point
circle
(iii) If l (TA) < l (CA) ; no real circle
E l
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 33/70
Examples
Q. Tangents are drawn to the hyperbola x2 – y2 = a2
enclosing at an angle of 45°. Show that the2 2 2
+ 4a2 (a2 – y2) = 4a4.
Q Find common tangent to y2 = 8x and 3x2 y2 = 3
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 34/70
Q. Find common tangent to y = 8x and 3x –y = 3
Q Find Tangent to passing through
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 35/70
Q. Find Tangent to passing through
(0, 4).
Q Prove that the two tangents drawn from any
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 36/70
Q. Prove that the two tangents drawn from any
point on the hyperbola x
2
– y
2
= a
2
– b
2
to the
ellipse make complementary angles
with the axes.
Chord Of Hyperbola
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 37/70
Chord Of Hyperbola
If (1) passes through (d, 0) then
Tangents And Normals
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 38/70
Tangents And Normals
(1) Certesian Tangent :
P t i T t
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 39/70
Parametric Tangent :
C t i N l
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 40/70
Certesian Normal :
P t i N l
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 41/70
Parametric Normal :
at (a secθ, b tanθ)
Q For the hyperbola x2 y2 a2 equations of the
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 42/70
Q. For the hyperbola x2 – y2 = a2, equations of the
normal becomes
Q
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 43/70
Q.
Q Find the equation to common tangent to the
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 44/70
Q. Find the equation to common tangent to the
hyperbolas
Q Perpendicular form the centre upon the tangent
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 45/70
Q. Perpendicular form the centre upon the tangent
and normal at any point of the hyperbola
meet them in Q and R. Find their loci.
* Chord of contact ; Chord with a given mid
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 46/70
point ; Pair of tangents
Q. From points on the circle x2 + y2 = a2 tangents
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 47/70
are drawn to the hyperbola x2 – y2 = a2; prove
that the locus of the middle points of the
chords of contact is the curve
(x – y ) = a (x + y )
Q. A point P moves such that the chord of contact
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 48/70
of the pair of tangents from P on the parabola
y2 = 4ax touches the rectangular hyperbola
x2 – y2 = c2. Show that the locus of P is the
ellipse
Q. Find the equation to the locus of the middle2 2
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 49/70
points of the chords of the hyperbola 2x2–3y2=1,
each of which makes an angle of 45° with the
x-axis.
Q. A tangent to the hyperbola
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 50/70
Q g yp
cut the ellipse 1 at P and Q. show that
the locus of the mid oints of P is
Q. Show that the mid points of focal chords of a
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 51/70
Q p
hyperbola 1 lie on another similar
h erbola.
Highlights
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 52/70
Highlights
H-1 Locus of the feet of the perpendicular drawn
any tangent is its auxiliary circle i.e. x2+y2 = a2
& the product of the feet of these2 2
H-2 The portion of the tangent between the point
f & h di i b d i h
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 53/70
of contact & the directrix subtends a right
angle at the corresponding focus.
H-3 The tangent & normal at any point of a
h b l bi t th l b t th f l
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 54/70
hyperbola bisect the angle between the focal
radii. This the reflection property of the
hyperbola as “An incoming light ray” aimed
towards one focus is reflected from the outersurface of the hyperbola towards the other
focus. It follows that if an ellipse and
hyperbola have the same foci, they cut at right
an les at an of their common oint.
Asymptotes
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 55/70
Definition : If the length of the perpendicular letfall from a point on a hyperbola to a straight line
ten s to zero as t e po nt on t e yper o a moves toinfinity along the hyperbola, then the straight line is
called the Asymptote of the Hyperbola.
To find the asymptote of the hyperbola :
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 56/70
Particular Case
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 57/70
When b = a the asymptotes of the rectangularhyperbola. x2 – y2 = a2 are, y = ± x which are at
.
Note
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 58/70
(i) Equilateral hyperbola ⇔
rectangular hyperbola(ii) If a hyperbola is equilateral then the conjugate
.
(iii) A hyperbola and its conjugate have the sameasymptote.
(iv) The equation of the pair of asymptotes differ thehyperbola & the conjugate hyperbola by the
same cons an on y.
(v) The asymptotes pass through the centre of the
hyperbola & the bisectors of the angles between
the asymptotes are the axes of the hyperbola.
(vi) The asymptotes of a hyperbola are the diagonals
of the rectangle formed by the lines drawn
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 59/70
of the rectangle formed by the lines drawn
through the extremities of each axis parallel tothe other axis.
(vii) Asymptotes are the tangent to the hyperbola
from the centre
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 60/70
from the centre.
(viii) A simple method to find the coordinates of the
centre of the hyperbola expressed as a general
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 61/70
centre of the hyperbola expressed as a general
equation of degree 2 should be remembered as :Let f(x, y) = 0 represents a hyperbola.
Find . Then the point of intersection of
gives the centre of the hyperbola.
Q. Find the asymptotes of the hyperbola,
3x2 – 5xy – 2y2 – 5x + 11y – 8 = 0
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 62/70
3x – 5xy – 2y – 5x + 11y – 8 = 0.
Also find the equation of the conjugate hyperbola.
Q. Find the equation to the hyperbola whose
asymptotes are the straight line 2x + 3y + 3 = 0
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 63/70
asymptotes are the straight line 2x + 3y + 3 = 0
and 3x + 4y + 5 = 0 and which passes throughthe point (1, -1). Also write the equation to the
conjugate hyperbola and the coordinates of itscentre.
2
Rectangular Hyperbola
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 64/70
(a) Equation is xy = c2
with parametricrepresentation x = ct, y = c/t, t ∈ R – {0}.
(b) Equation of a chord joining the point P(t1) &
Q(t2) is x + t1t2y = c (t1 + t2) with slope
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 65/70
Q( 2) 1 2y ( 1 2) p
(c) Equation of the tangent at P (x1, y1) is
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 66/70
(d) Equation of normal :
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 67/70
(e) Chord with a given middle points as (h, k) is
kx + hy = 2hk.
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 68/70
y
(f) Equation of the normal at P(t) is xt3–yt = c(t4–1).
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 69/70
(f) If a circle and the rectangular hyperbola xy = c2
meet in the four parametric points t1, t2, t3 & t4,
7/17/2019 Hyperbola Slides 475
http://slidepdf.com/reader/full/hyperbola-slides-475 70/70
then prove t1 t2 t3 t4 = 1