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Pure Maths/Hyperbola/p.1
Hyperbola
 focus : (ae, 0)

a
For the standard form of the equation of the ellipse, 
directrix : x =

e

or
Let P(x, y) be a variable point on the hyperbola.
Distance of (x, y) from the focus (ae, 0)
a
Distance of (x, y) from the directrix x =
e
or
y
P(x, y)
x
= e
Distance of (x, y) from the focus (- ae, 0)
a
Distance of (x, y) from the directrix x = e
(-a, 0)
x
x
F’(-ae,0)
= e

 x - ae

a

+ y2 = e2  - x
e 
2




x2 1 - e 2 + y2 = a 2 1 - e 2

x2
y2
= 1
a 2 a 2 e2 - 1

Principal
focus
x=a
a
e
x=-
e
x2
y2
= 1
2
2
a
b
L
y

x B(0,b)
x2
y2
= 1
a2
b2

(a, 0)
x
x
x
F(ae, 0)
0
Secondary
focus
where e > 1.
2
 focus : (- ae, 0)

 directrix : x = - a

e

x

where b2 = a 2 e 2 - 1
V’(-a,0)
V(a,0)
x
x
0
x
F(ae,0)
.
x B’(0,-b)
Consider
L’
PF - PF'
=

a 

 a 
e  x -  - e  x -  -  
e 

 e 

= 2a
∴The locus of a variable point moving in such a way that the absolute value of the difference of its
distances from 2 fixed points (foci) is a constant is a hyperbola.
Remarks
1.
V, V’ - vertices of the hyperbola ;
VV’ - transverse axis
(a)
(b)
x2 y2
= 1
32 22
y2 x2
= 1
32 22
;
BB’
origin 0
- centre of the hyperbola
- conjugate axis
 ________________________________________________
(c)
x2 y2
= 1
22 32
Pure Maths/Hyperbola/p.2
(d)
2.
y2
22
-
x2
32
= 1
The chord through the focus and perpendicular to the transverse axis - latus rectum (rectara)

(ae)2 y2
= 1
a2
b2
∴

b4
 y2 = b2 e 2 - 1 =
a2
2b 2
length of latus rectum =
a


2a 2 e 2 - 1
LL' 2b2
1
=

=
AF
a a(e - 1)
a 2  e - 1
= 2 (1 + e) > 4
3.
Point of intersection of the transverse axis and conjugate axis is called the centre of the hyperbola.
4.
Asymptotes of Hyperbola :
y
x2
y2
= 1
a2
b2
x2
y2
= 0
a2
b2
Equation of two asymptotes :

x
y
= 0
a
b

5.

and
x
y
+
= 0
a
b

b2 = a 2 e 2 - 1

∴
6.
x
y  x
y
x
 -   +  = 0
a
b  a
b
b=a
iff
e=
2
The equation of the hyperbola becomes
x2 - y2 = a 2 and such a hyperbola is called the
standard form of rectangular hyperbola whose asymptotes are perpendicular to each other.
y
Parametric Equation :
x P
(a) Trigonometric Parametric Equations
x2 y2
= 1
a 2 b2
 x = a sec
 
 y = b tan
a
0

x
x
where  is the eccentric angle and 0   < 2 .
Auxiliary
circle : x2 + y2 = a2
Pure Maths/Hyperbola/p.3
(b) Algebraic Parametric Equations
 x = a sec

 y = b tan

Let t = tan

2
Ellipse :

1. b2 = a 2 1 - e 2
2.

a 1 + t2
x =

1 - t2
;
y =
x2
y2
+
= 1
a2
b2

 x + my + n = 0 touches the ellipse iff
Hyperbola :
Equation of tangent at (x1 , y1 ) :
Equation of normal at (x1 , y1 ) :
2.
 x + my + n = 0 touches the hyperbola iff
_______________________________
3.
Equation of tangent with slope m :
y = mx 
6.
4.
5.
6.
Equation of chord with (h, k) as mid-point :
Locus of mid-point of a system of parallel
7. Equation of chord with (h, k) as mid-point :
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8. Locus of mid-point of a system of parallel
chords with common slope m :
chords with common slope m :
b2
y = x
a 2m
9.
Conjugate diameters :
m m1 =
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9.
b2
a2
10. Auxiliary circle :
x2 + y2 = a2
Director circle :
_________________________________
hx
ky
h2
k2
+
=
+
a2
b2
a2
b2
8.
Equation of tangent with slope m :
_________________________________
x2 + y2 = a 2 + b2
7.
Equation of normal at (x1 , y1 ) :
_________________________________
b2 + a 2m 2
Director circle :
Equation of tangent at (x1 , y1 ) :
________________________________
a2 x
b2 y
= a 2 - b2
x1
y1
5.

b2 = a 2 e 2 - 1
xx1
yy
+ 1 = 1
a2
b2
4.

x2
y2
= 1
a2
b2
1.
a 2 2 + b 2 m 2 = n 2
3.
2bt
1 - t2
Conjugate diameters :
__________________________________
10. Auxiliary circle :
_________________________________
Pure Maths/Hyperbola/p.4
Example
1.
Find the centre and asymptotes of the hyperbola
9x2 - 16y2 + 18x - 32y - 151 = 0 .
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2.
If the conic  x2 +  y2 = 1 is tangent to the straight lines 5x - y - 4 = 0 and 5x + 2y + 2 = 0,
find the values of  and  .
Hence, or otherwise, find the points of contact of the conic and the straight lines.
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y
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x
Pure Maths/Hyperbola/p.5
3.
Find the equation of the normal, with slope m, to the hyperbola
x2
a2
-
y2
b2
= 1
y
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x
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H.W. : W.43Ex.A. Q.1, 2
Rectangular Hyperbola
Consider the rectangular hyperbola xy = a’ ,
Case 1 : if a’ > 0 , let xy = c2 ,
y
To rotate x, y-axes to X, Y-axes by 45°.


 X
  = 
 Y



X
Y

  x
  
  y

xy = c2
45°
X 2 - Y 2 = ______________________________________
x
0
= ______________________________________
Case 2 : if a’ < 0 , let xy = - c2 ,
y
Y
To rotate x, y-axes to X, Y-axes by - 45°.


 X
  = 
 Y




  x
  
  y

xy = - c2
X 2 - Y 2 = ______________________________________
= ______________________________________
x
0
- 45°
X
Pure Maths/Hyperbola/p.6
For xy = c2 , the parametric equations are usually taken as :

 x = ct
 y = c

t

for any non-zero parameter t.
Example
1.
(a)
c

Show that the equation of the normal to the hyperbola xy = c2 at the point  ct ,  is

t
ct 4 - x t 3 + y t - c = 0 .
(b) If the normals at the points whose parameters are t1 , t2 , t3 , t4 are concurrent, show that the
concurrent point has coordinates given by x = c ( t1 + t2 + t3 + t4 ) and
1
1
1
1
y = c  +
+
+  .
t2
t3
t4 
 t1
(a) ________________________________________________________________________
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(b) _________________________________________________________________________
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Note :
If Pi are conormal points on the rectangular hyperbola xy = c2 with parameters ti (I = 1, 2, 3, 4),
then t1 t2 t3 t4 = -1 .
2.

The parametric coordinates of any point P on the rectangular hyperbola xy = c2 are  ct ,

c
.
t
Four points P1 , P2 , P3 and P4 on the hyperbola have parameters t1 , t2 , t3 and t4 respectively.
(a) Prove that P1 , P2 , P3 and P4 lie on a circle if and only if t1 t2 t3 t4 = 1 .
(b) State the relationship between the hyperbola and the circle through the points P1, P3 and P4
if t12 t3 t4 = 1 .
(c) Given a point P, with parameter t and t2  1, on the hyperbola. Show that there exist two circles
which touch the hyperbola at P and again at a second point. If the second point of contact are Q
for one circle and R for the other, show that QR passes through the centre of the hyperbola and
PQ is perpendicular to PR.
Pure Maths/Hyperbola/p.7
Hence, give a sketch of points P, Q, R and the hyperbola.
(a) ________________________________________________________________________
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Classwork
 1
Let P  t ,  be a point on the rectangular hyperbola xy = 1. The normal at P meets the hyperbola again
 t
at the point Q.
(a) Find the values of t if the tangents at P and Q are parallel.
(b) If the tangents at P and Q are not parallel, find the point of intersection of these two tangents, T.
Hence, find the equation of the locus of the point T.
(a)
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Pure Maths/Hyperbola/p.8
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(b)
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W.43 Ex.B Q.6, 9
H.W. W.43 Ex.B Q.1, 2, 7, 8, 10
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