Conic Sections
Table : 18.2
Conic Section : General
Definition
S. No.
The curves obtained by intersection of a plane and a double cone in
different orientation are called conic section.
1.
  0, h  0, a  b , e = 0
A circle
2.
Definitions of various important terms
  0, ab  h  0 , e = 1
A parabola
3.
  0, ab  h2  0 , e < 1
An ellipse
(1) Focus : The fixed point is called the focus of the conic-section.
(2) Directrix : The fixed straight line is called the directrix of the conic
section.
(3) Eccentricity : The constant ratio is called the eccentricity of the
conic section and is denoted by e.
(4) Axis : The straight line passing through the focus and
perpendicular to the directrix is called the axis of the conic section. A conic
is always symmetric about its axis.
(5) Vertex : The points of intersection of the conic section and the axis
are called vertices of conic section.
(6) Centre : The point which bisects every chord of the conic passing
through it, is called the centre of conic.
(7) Latus-rectum : The latus-rectum of a conic is the chord passing
through the focus and perpendicular to the axis.
(8) Double ordinate : The double ordinate of a conic is a chord
perpendicular to the axis.
(9) Focal chord : A chord passing through the focus of the conic is
called a focal chord.
(10) Focal distance : The distance of any point on the conic from the
focus is called the focal distance of the point.
4.
  0, ab  h  0 , e >1
A hyperbola
5.
  0, ab  h2  0 , a  b  0 , e  2
A rectangular
hyperbola
General equation of a conic section when its
focus, directrix and eccentricity are given
SP  ePM  SP 2  e 2 PM 2

2
Parabola
Definition
A parabola is the locus of a point which moves in a plane such that its
distance from a fixed point (i.e., focus) in the plane is always equal to its
distance from a fixed straight line (i.e., directrix) in the same plane.
Standard equation of the parabola
Let S be the focus, ZZ' be the directrix of the parabola and (x , y ) be
any point on parabola, then standard form of the parabola is y 2  4 ax .
Some other standard forms of parabola are
(i) Parabola opening to left i.e, y 2  4 ax
(ii) Parabola opening upwards i.e., x 2  4 ay
(iii) Parabola opening downwards i.e., x 2  4 ay
Some terms related to parabola
Y
M
Focal distance
Z
Z
P
Recognisation of conics
The equation of conics is represented by the general equation of
second degree ax 2  2hxy  by 2  2 gx  2 fy  c  0
......(i)
and discriminant of above equation is represented by  , where
  abc  2 fgh  af 2  bg 2  ch 2
Case I : When   0 .
In this case equation (i) represents the degenerate conic whose nature
is given in the following table.
Table : 18.1
  0 and ab  h  0
3.
  0 and ab  h 2  0
2
Focus
S(a,0)
Nature of conic
A pair of coincident straight
lines
A pair of intersecting straight
lines
A point
Case II : When   0 .
In this case equation (i) represents the non-degenerate conic whose
nature is given in the following table.
X
Axis
Latus rectum
F
L
(a,–2a)
Q
Y Table : 18.3 (h,2 ah )
(A   )
2.
Double ordinate
2
which is general equation of second degree.
  0 and ab  h 2  0`
x=a
Vertex
A
Thus the locus of (h, k ) is (x   )2  (y   )2  e 2 ( Ax 2 By 2C) ,
Condition
Focal chord
(a, 2a)L
M
S ( ,  )
F
Q
Directrix
2
S. No.
1.
(h,2 ah)
P(h, k)
x+a=0
 Ah  Bk  C 

(h   )2  (k   )2  e 2 

A 2  B 2 

Nature of conic
2
Z
Ax+By+C=0
Let S ( ,  ) be the focus, Ax  By  C  0 be
the directrix and e be the eccentricity of a conic. Let
P(h, k ) be any point on the conic. Let PM be the
perpendicular from P, on the directrix. Then by
definition,
Condition
Important
terms
y 2  4 ax
y 2  4 ax
x 2  4 ay
x 2  4 ay
Vertex
Focus
Directrix
(0, 0)
(a, 0)
x  a
(0, 0)
(–a, 0)
x a
(0, 0)
(0, a)
y  a
(0, 0)
(0, –a)
y=a
Axis
y 0
y 0
x 0
x 0
4a
4a
4a
4a
x a
ax
y a
ay
Latusrectum
Focal distance
P( x , y )
Special form of parabola (y – k)2 = 4a(x – h) = a
The equation of a parabola with its vertex at (h, k) and axis as parallel
to x-axis is (y  k )2  4 a(x  h) .
Y
Y
x=h–a
a
V
(h,k)
S
X
Directrix
X
O
Y
X
Conic Sections
If the vertex of the parabola is (p, q) and its axis is parallel to y-axis,
then the equation of the parabola is (x  p)2  4 b(y  q) .
Parametric equations of a parabola
Table : 18.4
y 2  4 ax
y 2  4 ax
x 2  4 ay
x 2  4 ay
Parametric
Co-ordinates
(at 2 ,2at)
(at 2 ,2at)
(2at, at 2 )
( 2at,  at 2 )
x  at 2
x  at 2
x  2at
x  2at ,
y  2at
y  2at
y  at
y  at 2
(2at, at 2 )
tx  y  at 2
x 2  4 ay
(2at,  at 2 )
tx  y  at 2
2
Equations of tangent of all other parabolas in slope form
Position of a point and a line with respect to a
parabola
(1) Position of a point with respect to a parabola : The point
P(x 1 , y1 ) lies outside, on or inside the parabola y 2  4 ax according as
y12  4 ax 1 , ,  0 .
P
x 2  4 ay
Equation of
parabolas
x  h  at 2 and y  k  2 at .
P
ty  x  at 2
Table : 18.7
The parametric equations of parabola (y  k )2  4 a(x  h) are
Y
(at 2 ,2at)
(3) Slope Form
Parabola
Parametric
Equations
y 2  4 ax
Point of contact in
terms of slope (m)
Condition of
Tangency
y 2  4 ax
 a 2a 
 2,

m m 
y  mx 
a
m
c
y 2  4 ax
2a 
 a
  2 ,

m
 m
y  mx 
a
m
c
x 2  4 ay
(2am, am2 )
y  mx  am2
c  am 2
x 2  4 ay
(2am,am2 )
y  mx  am2
c  am 2
a
m
a
m
Point of intersection of tangents at any two
points on the parabola
(1) The point of intersection of tangents at two points P(at12 , 2 at1 )
(on)
and Q(at 22 , 2 at 2 ) on the parabola y 2  4 ax is (at1 t 2 , a(t1  t 2 )) .
P(inside)
(Outside)
Equation of tangent
in terms of slope (m)
Y
X
2
(at1 , 2at1 )
P
(at t ,a(t +t ))
1
(2) Intersection of a line and a parabola: The line y  mx  c does
not intersect, touches or intersect a parabola y  4 ax , according as
a
c , ,  .
m
Condition of tangency : The line y  mx  c touches the parabola, if
c  a/m .
2
1
2
R
X
O
X
2
Equations of tangent in different forms
Q(at22 ,2at2 )
(2) The locus of the point of intersection
of tangents to the parabola
Y
y  4 ax which meet at an angle  is
2
(x  a)2 tan 2   y 2  4 ax .
(3) Director circle: The locus of the point of intersection of
perpendicular tangents to a conic is known as its director circle. The
director circle of a parabola is its directrix.
(4) The tangents to the parabola y 2  4 ax at P(at12 , 2 at1 ) and
(1) Point Form
Q(at22 , 2 at2 ) intersect at R. Then the area of triangle
Table : 18.5
Equations of tangent of all other standard parabolas at
(x1, y1)
Tangent at (x1, y1)
Equation of parabola
y2 = 4ax
yy1 = 2a (x + x1)
y 2  4 ax
yy 1  2a(x  x 1 )
x  4 ay
xx 1  2a(y  y 1 )
x 2  4 ay
xx 1  2a(y  y1 )
2
is
Equation of pair of tangents from a point to a
parabola
The combined equation of the pair of the tangents drawn from a point
to a parabola is SS '  T 2 , where S  y 2  4 ax ; S '  y 12  4 ax 1 and
T  yy 1  2a(x  x1 ) .
Y
(2) Parametric form
Table : 18.6
Q
Equations of tangent of all other standard parabolas at 't'
Tangent at 't'
Parametric
coEquations of parabolas
ordinates 't'
y 2  4 ax
PQR
1 2
a (t1  t2 )3 .
2
at ,2at
2
ty  x  at 2
X
(x1,y1)P
O
X
R
The two tangents can be drawn Yfrom
a point to a parabola. The two

tangent are real and distinct or coincident or imaginary according as the
given point lies outside, on or inside the parabola.
Conic Sections
If the normal at the point P(at 12 ,2at1 ) meets the parabola y 2  4 ax
Equations of normal in different forms
again at (at 22 ,2at 2 ) ,
(1) Point form
Table : 18.8
then t 2  t 1 
Equation of normals of all other standard
parabolas at (x1, y1)
Equation of parabola
Normal at (x1, y1)
y
y2 = 4ax
y – y1 =
1
Co-normal points
Y
The points on the curve at
Q
which the normals pass through a
common point are called co-normal
P(x1,y1)
points. Q, R, S are co-normal points. X
X
O
The co- normal points are also called
R
the feet of the normals.
Properties of co-normal points
S
Y
(1) Three normals can be drawn
from a point to a parabola.
(2) The algebraic sum of the slopes of three concurrent normals is
zero.
(3) The sum of the ordinates of the co-normal points is zero.
(4) The centroid of the triangle formed by the co-normal points lies
on the axis of the parabola.
(5) The centroid of a triangle formed by joining the foots of the
normal of the parabola lies on its axis and is given by
 am 12  am 22  am 32 2am 1  2am 2  2am 3 


,


3
3


(x – x1)
2a
y1
(x  x 1 )
2a
2a
y  y1  
(x  x 1 )
x1
y  y1 
y  4 ax
2
x 2  4 ay
y  y1 
x 2  4 ay
2a
(x  x 1 )
x1
(2) Parametric form
Table : 18.9
Equations of normal of all other standard parabola at 't'
Equations of
Parametric coNormals at 't
parabolas
ordinates
2
2
y = 4ax
(at , 2at)
y + tx = 2at + at3
y 2  4 ax
(at 2 ,2at)
y  tx  2at  at 3
x 2  4 ay
(2at, at 2 )
x  ty  2at  at 3
x 2  4 ay
(2at,at 2 )
x  ty  2at  at 3
 am 12  am 22  am 32 
=
,0  .


3


(6) If three normals drawn to any parabola y 2  4 ax from a given
point (h, k) be real, then h  2a for a  1 , normals drawn to the
parabola y 2  4 x from any point (h, k) are real, if h  2 .
(7) Out of these three at least one is real, as imaginary normals will
always occur in pairs.
(3) Slope form
Table : 18.10
Equations of normal, point of contact, and condition of normality in terms of
slope (m)
Equations of
Point of
Equations of normal in
Condition of
parabola
contact in
terms of slope (m)
normality
terms of slope
(m)
c  2am  am3
y  mx  2am  am3
c  2am  am 3
(am2 ,2am)
y 2  4 ax
(am2 ,2am)
x  4 ay
 2a a 
, 2

 m m 
y  mx  2a 
a
m2
c  2a 
x  4 ay
a 
 2a
 , 2 
m
m


y  mx  2a 
a
m2
c  2a 
2
Equation of the chord of contact of tangents to a
parabola
y  mx  2am  am3
y 2  4 ax
2
a
m2
a
m2
Point of intersection of normals at any two
points on the parabola
The point of intersection of normals at any
P(at12 , 2 at1 ) and
Q(at 22 , 2 at 2 ) on
the
parabola
two points
y 2  4 ax is
R [2a  a(t 12  t 22  t 1 t 2 ),  at 1 t 2 (t 1  t 2 )]
Q
X
Chord of
contact
(x1,y1)P
O
Q
2
, 2 at 2 ) 't ' if normal at 't '
Relation between 't(1at' 2 and
2
1
meets the parabola again at 't2'
Y (at12 , 2 at1 )
P
X
A
Y
X
R
Y'
y 2  4 ax is yy 1  2a(x  x1 ) .
Equation of the chord of the parabola which is
bisected at a given point
The equation of the chord at the parabola y 2  4 ax
bisected at the point (x 1 , y 1 ) is given by T  S 1,
(x2,y2)Q
P(x1,y1)
y12
 4ax1 .
(x3,y3)R
Equation of the chord joining any two points on
the parabola
X
A
X
Y
where T  yy 1  2a(x  x1 ) and S 1 
R
Y
Let PQ and PR be tangents
to the parabola y 2  4 ax drawn
from any external point P(x1 , y1 )
then QR is called the chord of
contact
of
the
parabola
y 2  4 ax .
The chord of contact of
tangents drawn from a point
to
the
parabola
(x 1 , y 1 )
i.e., yy 1  2a(x  x1 )  y12  4 ax 1
Y (at12 , 2 at1 )
P
X
2
.
t1
Q
(at 22 , 2 at 2 )
Conic Sections
Let P(at12 ,2at1 ), Q(at22, ,2at2 ) be any two points on the parabola
Equation of polar : Equation of polar of the point (x 1 , y 1 ) with
y  4 ax . Then, the equation of the chord joining these points is,
2
y  2at 1 
x  at 12 or y(t1  t2 )  2 x  2at1t2 .
t1  t 2
(1) Condition for the chord joining points having parameters t and t to
be a focal chord: If the chord joining points (at 12 ,2at 1 ) and (at 22 , 2 at 2 ) on
respect to parabola y 2  4 ax is same as chord of contact and is given by
yy 1  2a(x  x1 ) .
2


1
Q
(h,k)
T
Q
2
the parabola passes through its focus, then (a,0 ) satisfies the equation
y(t1  t2 )  2 x  2at1t2
(h,k) T
Pole
P
(x1,y1)
R
T
Polar
 n 2 am 
respect to the parabola y 2  4 ax is  ,
.
l 
l
Diameter of a parabola
Definition
The locus of the middle points
of a system of parallel chords is
called a diameter and in case of a
parabola this diameter is shown to
be a straight line which is parallel
to the axis of the parabola.
The equation of the diameter
bisecting chords of the parabola
P(x1,y1)
Y
X
y=mx+c
R(h,k)
A
X
Ellipse
An ellipse is the locus of a point which moves in such a way that its
distance from a fixed point is in constant ratio (<1) to its distance from a
fixed line. The fixed point is called the focus and fixed line is called the
directrix and the constant ratio is called the eccentricity of the ellipse,
denoted by (e).
Standard equation of the ellipse
Q(x2,y2)
Let S be the focus, ZM be the directrix of the ellipse and P(x , y ) is
P
Since e  1 , therefore a 2 (1  e 2 )  a 2  b 2  a 2 .
Y
X
X
y2=4ax
 y . Then,
(1) Length of tangent  PT  PN cosec   y 1 cosec 
(2) Length of normal  PG  PN cosec (90   )  y 1 sec 
o
(3) Length of subtangent  TN  PN cot   y 1 cot 
Y
(0,b)
M
p(x,y)
M Z
Z
C
S(ae,0) A
A S(–ae,0)
(a,0)
(–a,0)
(0,–b) B
Y
x=–a/e
x=a/e
The other form of equation of ellipse is
a 2  b 2 (1  e 2 ) i.e ., a  b .
(4) Length of subnormal  NG  PN cot(90   )  y1 tan 
y=b/e
2a
 m , [Slope of tangent at P(x, y)]
y1
X
(1) Length of tangent at (at 2 ,2at)  2at cosec 
B
(–a,0)
 2at (1  cot 2  )  2at 1  t 2
(2) Length of normal at (at 2 , 2 at)  2 at sec 
 2 at (1  tan 2  )  2a t 2  t 2 tan 2 
 2a (t 2  1)
y=–b/e
(4) Length of subnormal at (at ,2at)  2at tan   2a .
S
B
X
(a,0)
A(0,– b)
Z
K
Ellipse
 x 2

y2
 2  2  1
 a

b
Imp.
terms
Pole and Polar
The locus of the point of intersection of the tangents to the parabola
at the ends of a chord drawn from a fixed point P is called the polar of
point P and the point P is called the pole of the polar.
C (0,0)
Difference between both ellipses will be clear from the following table :
Table Y: 18.11
(3) Length of subtangent at (at 2 ,2at)  2at cot   2at 2
2
x 2 y2

 1 , where,
y2 b2
K
Z
A(0,b)
S
Length of Tangent, Subtangent, Normal and
subnormal to y2 = 4ax at (at2, 2at)
X
Y
o
where , tan  
Directrix
 S(a,0)
A
N G(x1,2a,0)
(x1,0)
(0,be)
y 2  4 ax . Let the tangent and yy1=2a(x+x1)
normal at P(x1 , y1 ) meet the
X
axis of parabola at T and G (–x1,0)T
respectively, and tangent at
P(x1 , y1 ) makes angle  with
the positive direction of x-axis.
A(0, 0) is the vertex of the parabola and PN
b 2  a 2 (1  e 2 ) .
(x1,y1)
Y
(0,–be)
parabola
x 2 y2

 1 , where
a2 b 2
any point on the ellipse, then by definition
Directrix
Y
Length of tangent, subtangent, normal, subnormal
the
Q
Diameter
y 2  4 ax of slope m is y  2a / m .
Let
Polar
R
R pole of the line lx  myT n  0 with
Coordinates of pole : The
1
.
t1
(2) Length of the focal chord: The length of a focal chord having
parameters t1 and t 2 for its end points is a(t2  t1 )2 .
 0  2a  2at1 t 2  t1 t 2  1 or t2  
Q
R
Pole
P(x1,y1)
Centre
Vertices
For a > b
(0, 0)
For b > a
(0, 0)
(a, 0)
(0,  b)
Conic Sections
Length of major axis
Length of minor axis
Foci
2a
2b
2b
2a
(ae, 0)
(0, be )
y  b / e
Equation of directrices
x  a / e
Relation in a, b and e
b  a (1  e )
a  b 2 (1  e 2 )
Length of latus rectum
2b 2
a
2a 2
b
2
Ends of latus-rectum
2
2

b2
  ae,

a

2




Equations of tangent in different forms
(1) Point form: The equation of the tangent to the ellipse
xx
yy
x 2 y2

 1 at the point (x 1 , y 1 ) is 21  21  1 .
a
b
a2 b 2
(2) Slope form: If the line y  mx  c touches the ellipse
 a2



 b ,be 


(a cos, b sin  )
Parametric equations
x2 y2

 1 in two
a2 b 2
distinct points if a 2 m 2  b 2  c 2 , in one point if c 2  a 2 m 2  b 2 and
does not intersect if a 2 m 2  b 2  c 2 .
The line y  mx  c intersects the ellipse
(a cos, b sin  )
x 2 y

1,
a2 b 2
(0    2 )
Focal radii
Sum of focal radii
SP  S ' P 
Distance between foci
Distance between
directrices
Tangents at the vertices
SP  a  ex 1
SP  b  ey 1
S ' P  a  ex 1
S ' P  b  ey 1
2a
2b
2ae
2be
2a/e
2b/e
x = –a, x = a
y = b, y = – b
  a 2m
x 2 y
 b2
 2  1 at 
,
2
 a 2m 2  b 2
a
b
a 2m 2  b 2

(3) Parametric form: The equation of
x
y
(a cos  , b sin  ) is
cos   sin   1 .
a
b
Let the equation of ellipse in standard form will be given by
x
y2
 2 1.
2
a
b
Then the equation of ellipse in the parametric form will be given by
x  a cos  , y  b sin  , where  is the eccentric angle whose value vary
from 0    2 . Therefore coordinate of any point P on the ellipse will
be given by (a cos  , b sin  ) .
Equation of pair of tangents SS = T
(x  h)2 (y  k )2

1.
a2
b2
of
the
curve
is
(lx  my  n)2
a2
(mx  ly  p)
 1 , where lx  my  n  0 and mx  ly  p  0
b2
lx  my  n
 X,
are perpendicular lines, then we substitute
l2  m 2
mx  ly  p
 Y , to put the equation in the standard form.
l2  m 2
2

2
Y
2
x
y

 1 is the
a2 b 2
equation of an ellipse. The point
lies outside, on or inside the
ellipse
as
if
2
2
x
y
S 1  12  12  1 , ,  0
a
b
P(outside)
P(on)
and let
X
P(inside)
C
Intersection of a line and an ellipse
Y
line

.


tangent at any point
2
Pair of tangents: The equation of pair of tangents PA and PB is
x2
where S 
a
T 

2
x 12
a2
xx 1
y2
b2

1
y 12
b2
yy 1
1
A
(x1,y1)P
B
 2 1
a2
b
Director circle: The director circle is the locus of points from which
perpendicular tangents are drawn to the ellipse.
Hence locus of
P(x1 , y1 ) i.e., equation of director circle is
x y a b .
2
2
2
2
Equations of normal in different forms
(1) Point form: The equation of the normal at (x 1 , y 1 ) to the ellipse
2
x
y2
a2 x b 2 y
 2  1 is

 a2  b 2 .
2
x1
y1
a
b
(2) Parametric form: The equation of the normal to the ellipse
x 2 y2

 1 at (a cos  , b sin  ) is ax sec   by cosec   a 2  b 2 .
a2 b 2
Position of a point with respect to an ellipse
Let P(x1 , y1 ) be any point
straight
SS 1  T 2 ,
S1 
(1) If the centre of the ellipse is at point (h, k ) and the directions of
the axes are parallel to the coordinate axes, then its equation is
equation
the
1
Special forms of an ellipse
the
Hence,
Points of contact: Line y  mx  a 2m 2  b 2 touches the ellipse
2
If
c 2  a2m 2  b 2 .
y  mx  a 2m 2  b 2 always represents the tangents to the ellipse.
Parametric form of the ellipse
(2)
then
X
(3) Slope form: If m is the slope of the normal to the ellipse
then the equation of normal is y  mx 
m(a 2  b 2 )
a 2  b 2m 2
The co-ordinates of the point of contact are

 a2
 mb 2

,
 a 2  b 2m 2
a 2  b 2m 2


 .


.
x 2 y2

1,
a2 b 2
Conic Sections
Auxiliary circle
The circle described on the
major axis of an ellipse as diameter is
called an auxiliary circle of the ellipse.
Y
xx 1 yy 1
 2  1 , i.e., T  0 .
a2
b
Coordinates of pole: The pole of the line lx  my  n  0 with
Q
P(x,y)
x 2 y2
If 2  2  1 is an ellipse,
a
b
then
its
auxiliary
circle
is
x 2  y 2  a2 .
Eccentric angle of a point: Let P
X
X
x2
a
Y
2

y2
b2
1
x 2 y2

 1 . Draw PM perpendicular from P
a2 b 2
on the major axis of the ellipse and produce MP to meet the auxiliary circle
in Q. Join CQ. The angle XCQ   is called the eccentric angle of the
point P on the ellipse.
Note that the angle XCP is not the eccentric angle of point P.
be any point on the ellipse
If PQ and PR be the tangents through point P(x1 , y1 ) to the ellipse
Y
Q
X’
(x1,y1)P
X
C
R
Y
The equation of the chord of the ellipse
x 2 y2

 1, whose mid
a2 b 2
point be (x1 , y1 ) is T  S 1
P (x1,y1)
x12 y12

1 .
a2 b 2
R(x3,y3)
Equation of the chord joining two points on an ellipse
The equation of the chord joining two points having eccentric angles
x 2 y2

 1 is
a2 b 2
x
    y
   
   
cos 
  sin
  cos 
.
a
 2  b
 2 
 2 
Let P(x1 , y1 ) be any point inside or outside the ellipse. A chord
through P intersects the ellipse at A and B respectively. If tangents to the
ellipse at A and B meet at Q(h,k) then locus of Q is called polar of P with
respect to ellipse and point P is called pole.
A
Q(h,k)
A
A
Polar
P(x1,y1)
B
T
Pole
B
B
Pole
P(x1,y1)
Polar
B
A
Q
Equation of polar : Equation of polar of the point (x 1 , y 1 ) with
respect to ellipse
x 2 y2

 1 is given by
a2 b 2
2
y=mx+c
X
x2
a2
2
X

y2
b2
1
y
b
2
2
x
a m
x
y
b
Y
 2  1 is y   2 x ,
2
a m
a
b
which is passing through (0, 0).
Conjugate diameter : Two diameters of an ellipse are said to be
conjugate diameter if each bisects all chords parallel to the other. The
coordinates of the four extremities of two conjugate diameters are
P(a cos , b sin ); P(a cos ,b sin )
Y
Q(a sin , b cos ); Q(a sin ,  b cos )
Q
B
A
90°

X
P
X
C
 b2
ellipse, then m1m 2  2 .
P
Q
A
B
a
(1) Properties of diameters : (i) The tangent at the extremity of any
diameter is parallel to the chords it bisects or parallel Yto the conjugate
diameter.
(ii) The tangent at the ends of any chord meet on the diameter which
bisects the chord.
(2) Properties of conjugate diameters : (i) The eccentric angles of the
ends of a pair of conjugate diameters (a cos ' , b sin  ' )
of an ellipse differ by a right angle, i.e.,
P(a cos, b sin  )
D

   .
A
A
2
C
Pole and Polar
Q(h,k)
Y
Definition : A line through the
centre of an ellipse is called a
diameter of the ellipse.
The equation of the diameter
bisecting the chords (y  mx  c)
of slope m of the ellipse
If y  m1 x and y  m2 x
be two conjugate diameters of an
Q(x2,y2)
xx
yy
where T  21  21  1 ,
a
b
 and  on the ellipse
l2 x  m2y  n2  0 , then the pole of the second line will lie on the first
and such lines are said to be conjugate lines.
(3) Pole of a given line is same as point of intersection of tangents at
its extremities.
2
Equation of chord with mid point (x1, y1)
S1 
of Q(x 2 , y 2 ) goes through P(x1 , y1 ) and such points are said to be
conjugate points.
(2) If the pole of a line l1 x  m 1 y  n1  0 lies on the another line
Diameter of the ellipse
Chord of contact
x 2 y2
then the

 1,
a2 b 2
equation of the chord of
contact
QR
is
xx 1 yy 1
 2  1 or T  0
a2
b
at ( x 1 , y 1 ) .
  a 2l  b 2m 
x 2 y2
.
,
 2  1 is P 
2
n 
a
b
 n
Properties of pole and polar
(1) If the polar of P(x1 , y1 ) passes through Q(x 2 , y 2 ) , then the polar
respect to ellipse

C M
(ii) The sum of the squares of any
P
D
two conjugate semi-diameters of an
ellipse is constant and equal to the sum of the squares of the semi axes of
the ellipse i.e., CP 2  CD 2  a 2  b 2 .
(iii) The product of the focal
P(a cos, b sin  )
D
distances of a point on an ellipse is equal
to the square of the semi-diameter which
is conjugate to the diameter through the
C
S
S
point i.e., SP .S P  CD 2 .
P
D
(iv) The tangents at the extremities
of a pair of conjugate diameters form a parallelogram whose area is
constant and equal to product of the axes. i.e., Area of parallelogram
 (2 a)(2b )
= Area of rectangle contained under major and minor axes.
Y
M
R
Q
P
D
R
C
D
P
Q
X
Conic Sections
Difference between both hyperbolas will be clear from the following table :
Table : 18.12
(v) The polar of any point with respect to ellipse is parallel to the diameter
to the one on which the point lies. Hence obtain the equation of the chord
whose mid point is (x 1, y1 ) , i.e. chord is T  S 1 .
(3) Equi-conjugate diameters: Two conjugate diameters are called
equi-conjugate, if their lengths are equal i.e., (CP )  (CD) .
2
 (CP )  (CD) 
2
2
Imp.
terms
Equation of directrices
ellipse
P(x1,y1)
x 2 y2

 1 is
a2 b 2
X
C
B D
A
Length of latus rectum
Parametric
co-ordinates
X
Y
x 2 y2

 1 is
a2 b 2

b2  b2
BD  CD  CB  x 1   x 1  2 x 1   2 x 1  (1  e 2 )x 1 .
a

 a
P(x1, y1 ) to the ellipse
Hyperbola
y  b / e
 a2  b 2
e  
2
 b




2a 2 / b
(b sec , a tan  )
0    2
SP  ex 1  a
SP  ey 1  b
S P  ex 1  a
S P  ey 1  b
2a
2b
x  a, x  a
y  b, y  b
y0
x 0
x 0
y 0
Difference of focal
radii (S P  SP )
Tangents at the
vertices
Equation of the
transverse axis
Equation of the
conjugate axis
Special form of hyperbola
If the centre of hyperbola is (h, k) and axes are parallel to the co-
Definition
A hyperbola is the locus of a point in a plane which moves in the
plane in such a way that the ratio of its distance from a fixed point in the
same plane to its distance from a fixed line is always constant which is
always greater than unity.
ordinate axes, then its equation is
Let S be the focus, ZM be the directrix and e be the eccentricity of the
x 2 y2
hyperbola, then by definition, 2  2  1 , where b 2  a 2 (e 2  1) .
a
b
L1
(–a,0) A Z C
Z A (a, 0)
Directrix
L1
Conjugate hyperbola
Directrix
(–ae,0)S
P L
M
M
Q
S
S
Y
b2
1 .
a
2

y2
b2
 1 be the hyperbola, then equation of the auxiliary
circle is x 2  y 2  a 2 .
Q
X
Q

(– a,0)A (0,0)C
90o
(x,y)
P
N
A(a,0)
X
Y
Parametric equations of hyperbola
The equations x  a sec  and y  b tan  are known as the
(0, be)
(0,–be)
x2
(ae,0)
Y
X
Let
X
x=–a/e and
x=a/e
Y conjugate
The hyperbola whose transverse
axis are respectively the
conjugate and transverse axis of a given hyperbola is called conjugate
hyperbola of the given hyperbola.
Z B(0,b) y= b/e
C
y= –b/e
Z
B (0,–b)
a2
(y  k ) 2
Y
S
L

Let QCN   . Here P and Q are the corresponding points on the
hyperbola and the auxiliary circle (0    2 ) .
Y
(x,y)
(x  h) 2
Auxiliary circle of hyperbola
Standard equation of the hyperbola
X
x  a / e
2b 2 / a
(a sec  , b tan  )
0    2
Focal radii
2
a
DA  CA  CD 
 x1 .
x1
Length of sub-normal at
(0, 0)
2a
2b
( ae, 0)





2
or
y2
 1
b2
(0, 0)
2b
2a
(0,  be)
 a2  b 2
e  
2
 a
Eccentricity
x2
y2
 2 1
2
a
b
x2
a
Subtangent and subnormal
Let the tangent and normal at P(x1 , y1 ) meet the x-axis at A and
Y
B respectively. Length of
A
subtangent at P(x1 , y1 ) to the
2
x
y

1
a2 b2
Centre
Length of transverse axis
Length of conjugate axis
Foci
(a 2  b 2 )
for equi-conjugate diameters.
2

Hyperbola
x 2 y2

1.
a2 b 2
(a sec , b tan ) lies on the hyperbola for all values of  .
parametric
X
equations
of
the
hyperbola
Position of a point with respect to a hyperbola
This
Conic Sections
Let the hyperbola be
Then
x 2 y2

1.
a2 b 2
the
P(x1 , y1 ) will lie inside, on or outside the hyperbola
x2 y2
x 12 y12
according
as


1

 1 is positive, zero or negative.
a2 b 2
a2 b 2
Y
A C
a2

y2
b2
 1 at (x 1 , y1 ) is
x 2 y2
The straight line y  mx  c will cut the hyperbola 2  2  1 in
a
b
two points may be real, coincident or imaginary according as
x2
a2
x2
x2 y2

 1 , then c 2  a 2m 2  b 2 .
a2 b 2

y2
b2

y2
2
xx
yy
x
y

 1 at (x1 , y1 ) is 21  21  1 .
a
b
a2 b 2
(2) Parametric form : The equation of tangent to the hyperbola
x 2 y2
x
y

 1 at (a sec , b tan  ) is sec   tan   1 .
a
b
a2 b 2
(3) Slope form : The equations of tangents of slope m to the hyperbola
x2 y2

 1 are y  mx  a 2m 2  b 2 and the co-ordinates of points
a2 b 2

a 2m


a 2m 2  b 2
,

.


b2
a 2m 2  b 2
x2
a2

y2
b2
 1 is
m (a 2  b 2 )
2
 1 , then c  
m (a 2  b 2 )
a2
a b m
2
2

.

a b m 
2
2
x2
Let PQ and PR be tangents to the hyperbola
a2

y2
b2
 1 drawn
from any external point P (x 1 , y 1 ) .
T 0,
xx 1
a
2

yy 1
b2
 1 or
Y
X
2
x
y
 2  1 then a pair
2
a
b
The equation of pair of tangents PQ and PR is SS 1  T 2
Y
A C P
(x1,y1)
Q
X
A C P A
(x1,y1)
R
X
A
R
Y
Equation of the chord of the hyperbola whose
mid point (x1, y1) is given
(h,k)T
Equation of the chord of the hyperbola
given point (x 1 , y1 ) is
y12
2
x
y
xx
yy

 1 , S1  2 
 1, T  21  21  1
a2 b 2
a
b
a
b
Director circle : The director circle is the locus of points from which
perpendicular tangents are drawn to the given hyperbola. The equation of
where, S 
2
Equation of chord of contact of tangents drawn
from a point to a hyperbola
of tangents PQ, PR can be drawn to it from P.
Y x 2
1
2
Q
If P(x1 , y1 ) be any point outside the hyperbola
2
,
mb 2
,
Then equation of chord of contact QR is
2
2
m 2 (a 2  b 2 ) 2
or c 2 
Equation of pair of tangents
X
is
 1 in terms of the slope m of the normal is




(1) Point form : The equation of the tangent to the hyperbola
of contacts are  
1
a
b
(a 2  m 2 b 2 )
a 2  m 2b 2
which is condition of normality.
(5) Points of contact : Co-ordinates of points of contact are
2
Equations of tangent in different forms
2
b2
.
a 2  b 2m 2
(4) Condition for normality : If y  mx  c is the normal of
2
Condition of tangency : If straight line y  mx  c touches the
hyperbola
a2
y2
a2 x b 2y

 a2  b 2 .
x1
y1
y  mx 
c , ,  a m  b .
2

ax cos   by cot  = a 2  b 2
(3) Slope form: The equation of the normal to the hyperbola
Y
Intersection of a line and a hyperbola
2
x2
hyperbola
(2) Parametric form: The equation of normal at (a sec  , b tan  ) to
X
the hyperbola
2
the
(1) Point form : The equation of normal to the hyperbola
P
A
of
Equations of normal in different forms
x2
P(inside)
X
circle
x 2  y 2  a2  b 2 .
P (outside)
(on)
director
xx 1
a
2

yy 1
b
2
1 =
x 12
a
2
x2
a2


y 12
b2
y2
b2
 1 , bisected at the
 1 i.e., T  S 1 .
Y
(x2,y2) Q
X
P
C
A
(x3,y3) R
Y
(x1,y1)
X
Conic Sections
]
Equation of the chord joining two points on the
hyperbola
The equation of the chord joining the points P(a sec 1 , b tan 1 )
and Q(a sec  2 , b tan  2 ) is
   2
x
cos  1
a
2

 y
   2
  sin 1
2
 b


   2
  cos  1
2



 .

The locus of the point of intersection of the tangents to the hyperbola
at A and B is called the polar of the given point P with respect to the
hyperbola and the point P is called the pole of the polar. The equation of
xx 1 yy 1
the required polar with (x 1 , y1 ) as its pole is
 2 1 .
a2
b
A
A
(h, k)Q
A
X
a2
y2
b2
a
Asymptotes of a hyperbola
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.
The equations of two asymptotes of the hyperbola
B
Polar
X
x2
A
y2
b2
 1 are
4 x 2  (4 h  k ) x  1  0 then the pole of the second line will lie on the
first and such lines are said to be conjugate lines.
(iii) Pole of a given line is same as point of intersection of tangents as
its extremities.
Diameter of the hyperbola
The locus of the middle points of a system of parallel chords of a
hyperbola is called a diameter
Y
and the point where the diameter
(x1,y1)P
intersects the hyperbola is called
the vertex of the diameter.
Let y  mx  c a system
X
X
C
of
parallel
chords
to
(x2,y2)Q
y2
R(h,k)
b2x
, which is passing through (0, 0).
a2m
Conjugate diameter : Two diameters are said to be conjugate when each
bisects all chords parallel to the others.
Subtangent and Subnormal of the hyperbola
y2
2
 1 is
x2
2

y2
Asymptotes
X
A
A
X
C
x2
B
a
Y x 2
2

y2
b2
1
y2
 2  1 differ the hyperbola and
(iv) The equation of the pair ofa 2asymptotes
b
the conjugate hyperbola by the same constant only i.e., Hyperbola –
Asymptotes = Asymptotes – Conjugated hyperbola or
 x2 y2
  x2 y2   x2 y2   x2 y2



 
 



1
 a2 b 2
  a 2  b 2    a 2  b 2    a 2  b 2  1 .

 
 
 

(v) The asymptotes pass through the centre of the hyperbola.
(vi) The bisectors of the angles between the asymptotes are the
coordinate axes.
(vii) The angle between the asymptotes of the hyperbola S  0 i.e.,
x2
a
2
y2
b
or 2 sec 1 e .
a
b
(viii) Asymptotes are equally inclined to the axes of the hyperbola.

2
 1 is 2 tan 1
Rectangular or equilateral hyperbola
Y
If y  m 1 x , y  m 2 x be conjugate diameters, then m 1 m 2 

B
polar of Q(x 2 , y 2 ) goes through P(x 1 , y1 ) and such points are said to
be conjugate points.
(ii) If the pole of a line lx  my  n  0 lies on the another line
diameter of the hyperbola is y 

0.
a
b2
a
b
(ii) When b  a i.e. the asymptotes of rectangular hyperbola
2
x  y 2  a 2 are y   x , which are at right angles.
(iii) A hyperbola and its conjugate hyperbola have the same
asymptotes.
Y
2
 a 2l b 2m 
.
,
 1 is (x1 , y1 )   
n 
 n

 1 for different
a2 b 2
chords then the equation of
2
b
x y
x or
 0.
a
a b
Some important points about asymptotes
(i) The combined equation of the asymptotes of the hyperbola
Properties of pole and polar
(i) If the polar of P(x 1 , y1 ) passes through Q(x 2 , y 2 ) , then the
x2
x2
a
Q B

Y
= b 2 x1  (e 2  1)x1 .
X
Pole of a given line: The pole of a given line lx  my  n  0 with
x2
(a 2  b 2 )
x1  x1
a2
y
Pole
P
(x1,y1)
X
2
B
B
B
N
A
C
Pole
P(x1,y1)
X
Q
X
Length of subnormal
BN  CB  CN 
(h,k) Q
Polar
(x1, y1)P
a2
.
AN  CN  CA  x 1 
x1
Pole and Polar
respect to the hyperbola
Let the tangent and normal at P(x 1 , y1 ) meet the x-axis at A and B
Y
respectively.
Length of subtangent
b2
a2
.
(1) Definition : A hyperbola whose asymptotes are at right angles to
each other is called a rectangular hyperbola. The eccentricity of rectangular
hyperbola is always 2 .
The general equation of second degree represents a rectangular
hyperbola if   0, h 2  ab and coefficient of x 2 + coefficient of y 2 =
0.
(2) Parametric co-ordinates of a point on the hyperbola XY = c : If t is
non–zero variable, the coordinates of any point on the rectangular
2
Conic Sections
hyperbola xy  c 2 can be written as (ct , c / t) . The point (ct, c / t) on
the hyperbola xy  c 2 is generally referred as the point ‘t’.
For rectangular hyperbola the coordinates of foci are (a 2 , 0) and
directrices are x  a 2 .
For rectangular hyperbola xy  c 2 , the coordinates of foci are
(c 2 ,  c 2 ) and directrices are x  y   c 2 .
(3) Equation of the chord joining points t and t : The equation of the


c
c 
chord joining two points  ct 1 ,  and  ct 2 ,  on the hyperbola
t1 
t2 


1
2
c c

c
t 2 t1
(x  ct1 )
xy  c is y  
t1 ct 2  ct1
2
 x  y t1t2  c (t1  t2 ) .
(4) Equation of tangent in different forms
(i) Point form : The equation of tangent at (x 1 , y1 ) to the hyperbola
xy  c 2 is xy 1  yx 1  2c 2 or
x
y

 2.
x1 y 1
 c
(ii) Parametric form : The equation of the tangent at  ct,  to the
 t
x
 yt  2 c .On replacing x 1 by ct and y 1 by
hyperbola xy  c 2 is
t
c
on the equation of the tangent at (x 1 , y1 )
t
x
i.e., xy 1  yx 1  2c 2 we get  yt  2 c .
t
Point of intersection of tangents at ' t1 ' and ' t2 ' is
 2ct1 t2
2c 


t t , t t .
 1 2 1 2
(5) Equation of the normal in different forms :
(i) Point form : The equation of the normal at (x 1 , y1 ) to the hyperbola
xy  c 2 is xx 1  yy 1  x 12  y 12 .
 c
(ii) Parametric form : The equation of the normal at  ct,  to the
 t
hyperbola xy  c 2 is xt 3  yt  ct 4  c  0 .
On replacing x 1 by ct and y 1 by c / t in the equation.
We obtain xx 1  yy 1  x 12  y 12 ,
yc
c2
xct 
 c 2 t 2  2  xt 3  yt  ct 4  c  0 .
t
t
This equation is a fourth degree in t. So, in general four normals can
be drawn from a point to the hyperbola xy  c 2 , and point of intersection
of normals at t1 and t2 is
 c {t1 t 2 (t12  t1 t 2  t 22 )  1} c {t13 t 23  (t12  t1 t 2  t 2 )} 

.
,


t1 t 2 (t1  t 2 )
t1 t 2 (t1  t 2 )


 The area of the triangle inscribed in the parabola y 2  4 ax is
1
(y1 ~ y 2 )(y 2 ~ y 3 )(y 3 ~ y1 ), where y1 , y 2 , y 3 are the ordinates
8a
of the vertices.
 The length of the side of an equilateral triangle inscribed in the
parabola y 2  4 ax is 8a 3 . (one angular point is at the vertex).
 y 2  4 a(x  a) is the equation of the parabola whose focus is the
origin and the axis is x-axis.
 y 2  4 a(x  a) is the equation of parabola whose axis is
xaxis and y-axis is directrix.
 x 2  4 a(y  a) is the equation of parabola whose focus is the
origin and the axis is y-axis.
 x 2  4 a(y  a) is the equation of parabola whose axis is
yaxis and the x-axis is directrix.
 The equation of the parabola whose vertex and focus are on x-axis
at a distance a and a' respectively from the origin is
y 2  4 (a'a)(x  a) .
 The equation of the parabola whose axis is parallel to x-axis is
x  Ay 2  By  C and y  Ax 2  Bx  C is a parabola with its axis
parallel to y-axis.
 If the straight line lx  my  n  0 touches the parabola
y 2  4 ax ,then l n  am 2 .

If the line x cos   y sin   p touches the parabola y 2  4 ax ,
p cos   a sin 2   0
then
and
point
of
contact
is
(a tan  ,2a tan  ) .
x
y

 1 touches the parabola y 2  4 a(x  b) ,
 If the line
l m
2
then m 2 (l  b)  al 2  0 .
 If the two parabolas y 2  4 x and x 2  4 y intersect at point P,
whose absiccae is not zero, then the tangent to each curve at P, make
complementary angle with the x-axis.
 Tangents at the extremities of any focal chord of a parabola meet
at right angle on the directrix.
 Area of the triangle formed by three points on a parabola is twice
the area of the triangle formed by the tangents at these points.

If the tangents at the points P and Q on a parabola meet in T, then
ST is the geometric mean between SP and SQ, i.e., ST 2  SP .SQ
 Tangent at one extremity of the focal chord of a parabola is parallel
to the normal at the other extremity.

The angle of intersection of two parabolas
x 2  4 by is given by tan 1

y 2  4 ax and
3 a1 / 3 b 1 / 3
.
2(a 2 / 3  b 2 / 3 )
The equation of the common tangents to the parabola y 2  4 ax
1
1
2
2
and x 2  4 by is a 3 x  b 3 y  a 3 b 3  0 .

The line lx  my  n  0 is a normal to the parabola y 2  4 ax ,
it al(l 2  2m 2 )  m 2 n  0 .

If the normals at points (at12 , 2 at ) and (at 22 , 2 at 2 ) on the parabola
y 2  4 ax meet on the parabola, then t1 t 2  2 .

If
the
normal
at
a
point
P(at 2 ,2at) to
the
parabola
Conic Sections
y 2  4 ax subtends a right angle at the vertex of the parabola then
t2  2 .

If the normal to a parabola y 2  4 ax , makes an angle  with the
1

axis, then it will cut the curve again at an angle tan 1  tan   .
2

 If the normal at two points P and Q of a parabola
y 2  4 ax intersect at a third point R on the curve. Then the product of
the ordinate of P and Q is 8 a 2 .
 The chord of contact joining the point of contact of two
perpendicular tangents always passes through focus.

If tangents are drawn from the point (x1 , y1 ) to the parabola
y  4 ax ,
then the length of their chord of
contact is
(at12 , 2 at1 )Q
2
1
| a|
(at 22 , 2 at 2 )R

 a  2a 
 by virtue of relation
extremity (at 22 , 2 at 2 ) becomes  2 ,
t
t1 
 1
t1 t 2  1 .
If one end of the focal chord of parabola is (at 2 ,2at) ,then other
2
1
a


end will be  2 ,  2 at  and length of chord  a t   .
t
t



The length of the chord joining two points ' t1 ' and ' t 2 ' on the
parabola y 2  4 ax is a(t1  t 2 ) (t1  t 2 ) 2  4 .
The length of intercept made by line y  mx  c between the
4
parabola y 2  4 ax is 2 a(1  m 2 )(a  mc ) .
m
 Locus of mid-points of all chords which is inscribed a right angle
on the vertices of parabola is parabola.


The focal chord of parabola y 2  4 ax making an angle  with
the x-axis is of length 4acosec 2 and perpendicular on it from the
vertex is asin.
 The length of a focal chord of a parabola varies inversely as the
square of its distance from the vertex.

If l1 and l2 are the length of segments of a focal chord of a
parabola, then its latus-rectum is
4 l1l2
.
l1  l2
x 2 y2

 1 , if a 2 l 2  b 2 m 2  n 2 .
a2 b 2

if
The line x cos   y sin   p touches the ellipse
a 2 cos 2   b 2 sin 2   p 2
x2
x 2 y2

1,
a2 b 2
and that point of contact is
2

y2

y2
y2
intercepted
If y  mx  c is the normal of
between
the
axes
is
x 2 y2

 1 , then condition of
a2 b 2
m 2 (a 2  b 2 )2
.
(a 2  b 2m 2 )
The straight line lx  my  n  0` is a normal to the ellipse
normality is c 2 
2
x 2 y2
a2
b 2  a 2  b 2 
if
.




1
,
a2 b 2
l2
m 2  n 2 
 Four normals can be drawn from a point to an ellipse.
Y
 If S be the focus and G be the point
Normal
Where the normal at P meets the axis of
P(x1,y1)
an ellipse, then SG  e.SP ,
and the tangent and normal
X
T
S C G S
at P bisect the external and
internal anglesbetween the focal distances of P.
X
Y
 Any point P of an ellipse is joined to the extremities
of the major
axis then the portion of a directrix intercepted by them subtends a right
angle at the corresponding focus.
 The equations to the normals at the end of the latera recta and
that each passes through an end of the minor axis, if e 4  e 2  1  0 .

The area of the triangle formed by the three points, on the ellipse
x 2 y2

 1 , whose eccentric angles
a2 b 2
       
  
2ab sin
 sin
 sin
.
 2   2 
 2 


are
If the point of intersection of the ellipses
x2
 The semi-latus rectum of the parabola y 2  4 ax is the harmonic
mean between the segments of any focal chord of the parabola.
lx  my  n  0 touches the ellipse
 The straight line
x2
1
a
b2
a2y 2  b 2 x 2  4 x 2y 2 .

If one extremity of a focal chord is (at 12 ,2at 1 ) ,then the other

 1 is (x 2  y 2 )2  a 2 x 2  b 2 y 2
a2 b 2
or r 2  a2 cos 2   b 2 sin2  . (In terms of polar coordinates)
 The locus of the mid points of the portion of the tangents to the
any tangent to the ellipse


A circle of radius r is concentric with the ellipse
 r2  b 2 
.
tan 1  2
2 
a r 
 The locus of the foot of the perpendicular drawn from centre upon
The area of the triangle formed by the tangents drawn from
(y 2  4 ax 1 ) 3 / 2
.
(x1 , y1 ) to y 2  4 ax and their chord of contact is 1
2a

x2
 1,
a2 b 2
then the common tangent is inclined to the major axis at an angle

ellipse
(x1,y1)P
(y 12  4 ax 1 )(y 12  4 a 2 ) .
 a 2 cos  b 2 sin  

.
,


p
p


2

y2
2
 ,  and
 is
x 2 y2

 1 and
a2 b 2
 1 be at the extremities of the conjugate diameters of the
former, then
a2

2

b2
2
2.
 The sum of the squares of the reciprocal of two perpendicular
diameters of an ellipse is constant.
 In an ellipse, the major axis bisects all chords parallel to the minor
axis and vice-versa, therefore major and minor axes of an ellipse are
conjugate diameters of the ellipse but they do not satisfy the condition
m 1 . m 2  b 2 / a 2 and are the only perpendicular conjugate diameters.
Conic Sections


The foci of a hyperbola and its conjugate are con-cyclic.
Two tangents can be drawn from an outside point to a hyperbola.
If the straight line lx  my  n  0 touches the hyperbola

x
2
a
2
y2
b2
 1 , then a 2 l 2  b 2 m 2  n 2 .
If the straight line x 2 cos   y sin   p touches the hyperbolas

x2
a

2

y2
b2
 1 , then a 2 cos 2   b 2 sin 2   p 2 .
If the line lx  my  n  0 will be normal to the hyperbola

x
y2
a2
b 2 (a 2  b 2 )2
.
 2  1 , then 2  2 
2
a
b
l
m
n2
 In general, four normals can be drawn to a hyperbola from any
point.
2
If ,  , 

are the eccentric angles of three points on the
2
x
y2
 2  1 , the normals at which are concurrent, then,
2
a
b
sin(   )  sin(   )  sin(  )  0 .
hyperbola.

The feet of the normals to
x2
a
2

y2
b2
 1 from (h, k ) lie on
a 2 y(x  h)  b 2 x (y  k )  0 .

The length of chord cut off by hyperbola
x2
a
line y  mx  c is
2

y2
 1 from the
b2
2ab [c 2  (a 2 m 2  b 2 )](1  m 2 )
(b 2  a 2 m 2 )
.
If the chord joining two points (a sec 1, b tan 1 ) and
(a sec  2 , b tan  2 ) passes through the focus of the hyperbola



1e
x 2 y2
 2  1 , then tan 1 tan 2 
.
2
2
2
1
e
a
b

If the polars of (x1 , y1 ) and (x 2 , y 2 ) with respect to the hyperbola
2
x
y2
x x
a4
 2  1 are at right angles, then 1 2  4  0 .
2
y1 y 2 b
a
b
 The parallelogram formed by the tangents at the extremities of
conjugate diameters of a hyperbola has its vertices lying on the
asymptotes and is of constant area.
 The product of length of perpendiculars drawn from any point on the
hyperbola
x 2 y2
a2b 2


1
to
the
asymptotes
is
.
a2 b 2
a2  b 2
 c
If the normal at  ct,  on the curve xy  c 2 meets the curve
 t
1
again in 't', then t    3 .
t

 A triangle has its vertices on a rectangular hyperbola; then the
orthocentre of the triangle also lies on the same hyperbola.
 All conics passing through the intersection of two rectangular
hyperbolas are themselves rectangular hyperbolas.
 An infinite number of triangles can be inscribed in the rectangular
hyperbola xy  c 2 whose all sides touch the parabola y 2  4 ax .