Conic Sections Class 11 Notes Maths Chapter 10

 Page 1


  
KEY POINTS 
? Circle, ellipse, parabola and hyperbola are curves which are 
obtained by intersection of a plane and cone in different 
positions 
? Circle:It is the set of all points in a plane that are equidistant 
from a fixed point in that plane 
? Equation of circle: (x – h)
2
 + (y – k)
2
 = r
2
 
Centre (h, k), radius = r 
? Parabola:It is the set of all points in a plane which are 
equidistant from a fixed point (focus) and a fixed line (directrix) 
in the plane. Fixed point does not lie on the line. 
Page 2


  
KEY POINTS 
? Circle, ellipse, parabola and hyperbola are curves which are 
obtained by intersection of a plane and cone in different 
positions 
? Circle:It is the set of all points in a plane that are equidistant 
from a fixed point in that plane 
? Equation of circle: (x – h)
2
 + (y – k)
2
 = r
2
 
Centre (h, k), radius = r 
? Parabola:It is the set of all points in a plane which are 
equidistant from a fixed point (focus) and a fixed line (directrix) 
in the plane. Fixed point does not lie on the line. 
  
 
 
Main facts about the Parabola 
Equation y
2
 = 4 a x y
2
 = –4 a x x
2
 = 4 a y x
2
 = –4 a y 
 (a > 0) a > 0 a > 0 a > 0 
 Right hand Left hand Upwards Downwards 
Axis y = 0 y = 0 x = 0 x = 0 
Directrix x + a = 0 x – a = 0 y + a = 0 y – a = 0 
Focus (a, 0) (–a, 0) (0, a) (0, –a) 
Length of 
latus–rectum 
4a 4a 4a 4a 
Equation of 
latus–rectum 
x – a = 0 x + a = 0 y – a = 0 y + a = 0 
? Latus Rectum:A chord through focus perpendicular to axis of 
parabola is called its latus rectum. 
O 
, 
X´ 
Y 
y  = 4ax 
2 
x 
 
+ 
 
a 
 
= 
 
0 
F(–a,0) 
O 
Y 
x 
 
– 
 
a 
 
= 
 
0 
X´ 
, 
y  = –4ax 
2 
F(a,0) 
O 
, 
Y 
x  = 4ay 
2 
F(0, a) 
X 
y + a = 0 
O 
Y 
x  = –4ay 
2 
F 
( 
0 
, 
 
– 
a 
) 
X 
y – a = 0 
Page 3


  
KEY POINTS 
? Circle, ellipse, parabola and hyperbola are curves which are 
obtained by intersection of a plane and cone in different 
positions 
? Circle:It is the set of all points in a plane that are equidistant 
from a fixed point in that plane 
? Equation of circle: (x – h)
2
 + (y – k)
2
 = r
2
 
Centre (h, k), radius = r 
? Parabola:It is the set of all points in a plane which are 
equidistant from a fixed point (focus) and a fixed line (directrix) 
in the plane. Fixed point does not lie on the line. 
  
 
 
Main facts about the Parabola 
Equation y
2
 = 4 a x y
2
 = –4 a x x
2
 = 4 a y x
2
 = –4 a y 
 (a > 0) a > 0 a > 0 a > 0 
 Right hand Left hand Upwards Downwards 
Axis y = 0 y = 0 x = 0 x = 0 
Directrix x + a = 0 x – a = 0 y + a = 0 y – a = 0 
Focus (a, 0) (–a, 0) (0, a) (0, –a) 
Length of 
latus–rectum 
4a 4a 4a 4a 
Equation of 
latus–rectum 
x – a = 0 x + a = 0 y – a = 0 y + a = 0 
? Latus Rectum:A chord through focus perpendicular to axis of 
parabola is called its latus rectum. 
O 
, 
X´ 
Y 
y  = 4ax 
2 
x 
 
+ 
 
a 
 
= 
 
0 
F(–a,0) 
O 
Y 
x 
 
– 
 
a 
 
= 
 
0 
X´ 
, 
y  = –4ax 
2 
F(a,0) 
O 
, 
Y 
x  = 4ay 
2 
F(0, a) 
X 
y + a = 0 
O 
Y 
x  = –4ay 
2 
F 
( 
0 
, 
 
– 
a 
) 
X 
y – a = 0 
  
? Ellipse: It is the set of points in a plane the sum of whose 
distances from two fixed points in the plane is a constant 
and is always greater than the distances between the 
fixed points 
 
Main facts about the ellipse 
 
Equation   
2 2
2 2
1
x y
a b
? ?
   
2 2
2 2
1
x y
b a
? ?
 
  a > 0, b > 0 a > 0, b > 
0
 
Centre (0,0) (0,0) 
Major axis lies along x–axis y–axis 
Length of major axis 2a 2a 
Length of minor axis 2b 2b 
Page 4


  
KEY POINTS 
? Circle, ellipse, parabola and hyperbola are curves which are 
obtained by intersection of a plane and cone in different 
positions 
? Circle:It is the set of all points in a plane that are equidistant 
from a fixed point in that plane 
? Equation of circle: (x – h)
2
 + (y – k)
2
 = r
2
 
Centre (h, k), radius = r 
? Parabola:It is the set of all points in a plane which are 
equidistant from a fixed point (focus) and a fixed line (directrix) 
in the plane. Fixed point does not lie on the line. 
  
 
 
Main facts about the Parabola 
Equation y
2
 = 4 a x y
2
 = –4 a x x
2
 = 4 a y x
2
 = –4 a y 
 (a > 0) a > 0 a > 0 a > 0 
 Right hand Left hand Upwards Downwards 
Axis y = 0 y = 0 x = 0 x = 0 
Directrix x + a = 0 x – a = 0 y + a = 0 y – a = 0 
Focus (a, 0) (–a, 0) (0, a) (0, –a) 
Length of 
latus–rectum 
4a 4a 4a 4a 
Equation of 
latus–rectum 
x – a = 0 x + a = 0 y – a = 0 y + a = 0 
? Latus Rectum:A chord through focus perpendicular to axis of 
parabola is called its latus rectum. 
O 
, 
X´ 
Y 
y  = 4ax 
2 
x 
 
+ 
 
a 
 
= 
 
0 
F(–a,0) 
O 
Y 
x 
 
– 
 
a 
 
= 
 
0 
X´ 
, 
y  = –4ax 
2 
F(a,0) 
O 
, 
Y 
x  = 4ay 
2 
F(0, a) 
X 
y + a = 0 
O 
Y 
x  = –4ay 
2 
F 
( 
0 
, 
 
– 
a 
) 
X 
y – a = 0 
  
? Ellipse: It is the set of points in a plane the sum of whose 
distances from two fixed points in the plane is a constant 
and is always greater than the distances between the 
fixed points 
 
Main facts about the ellipse 
 
Equation   
2 2
2 2
1
x y
a b
? ?
   
2 2
2 2
1
x y
b a
? ?
 
  a > 0, b > 0 a > 0, b > 
0
 
Centre (0,0) (0,0) 
Major axis lies along x–axis y–axis 
Length of major axis 2a 2a 
Length of minor axis 2b 2b 
  
Foci (–c, 0), (c, 0) (0, –c),(0, c) 
Vertices (–a, 0), (a, 0) (0, –a), (0, a) 
Eccentricity (e) 
Length of latus–rectum 
   
c
a
 
   
2
2 b
a
 
  
c
a
 
  
2
2 b
a
 
 
? If e = 0 for ellipse than b = a and equation of ellipse will be 
converted in equation of the circle its eq. will be x² + y² = a². it is 
called auxiliary circle. For auxiliary circle, diameter is equal to 
length of major axis and e = 0 
? Latus rectum:Chord through foci perpendicular to major axis 
called latus rectum. 
? Hyperbola:It is the set of all points in a plane, the differences of 
whose distance from two fixed points in the plane is a constant. 
 
Page 5


  
KEY POINTS 
? Circle, ellipse, parabola and hyperbola are curves which are 
obtained by intersection of a plane and cone in different 
positions 
? Circle:It is the set of all points in a plane that are equidistant 
from a fixed point in that plane 
? Equation of circle: (x – h)
2
 + (y – k)
2
 = r
2
 
Centre (h, k), radius = r 
? Parabola:It is the set of all points in a plane which are 
equidistant from a fixed point (focus) and a fixed line (directrix) 
in the plane. Fixed point does not lie on the line. 
  
 
 
Main facts about the Parabola 
Equation y
2
 = 4 a x y
2
 = –4 a x x
2
 = 4 a y x
2
 = –4 a y 
 (a > 0) a > 0 a > 0 a > 0 
 Right hand Left hand Upwards Downwards 
Axis y = 0 y = 0 x = 0 x = 0 
Directrix x + a = 0 x – a = 0 y + a = 0 y – a = 0 
Focus (a, 0) (–a, 0) (0, a) (0, –a) 
Length of 
latus–rectum 
4a 4a 4a 4a 
Equation of 
latus–rectum 
x – a = 0 x + a = 0 y – a = 0 y + a = 0 
? Latus Rectum:A chord through focus perpendicular to axis of 
parabola is called its latus rectum. 
O 
, 
X´ 
Y 
y  = 4ax 
2 
x 
 
+ 
 
a 
 
= 
 
0 
F(–a,0) 
O 
Y 
x 
 
– 
 
a 
 
= 
 
0 
X´ 
, 
y  = –4ax 
2 
F(a,0) 
O 
, 
Y 
x  = 4ay 
2 
F(0, a) 
X 
y + a = 0 
O 
Y 
x  = –4ay 
2 
F 
( 
0 
, 
 
– 
a 
) 
X 
y – a = 0 
  
? Ellipse: It is the set of points in a plane the sum of whose 
distances from two fixed points in the plane is a constant 
and is always greater than the distances between the 
fixed points 
 
Main facts about the ellipse 
 
Equation   
2 2
2 2
1
x y
a b
? ?
   
2 2
2 2
1
x y
b a
? ?
 
  a > 0, b > 0 a > 0, b > 
0
 
Centre (0,0) (0,0) 
Major axis lies along x–axis y–axis 
Length of major axis 2a 2a 
Length of minor axis 2b 2b 
  
Foci (–c, 0), (c, 0) (0, –c),(0, c) 
Vertices (–a, 0), (a, 0) (0, –a), (0, a) 
Eccentricity (e) 
Length of latus–rectum 
   
c
a
 
   
2
2 b
a
 
  
c
a
 
  
2
2 b
a
 
 
? If e = 0 for ellipse than b = a and equation of ellipse will be 
converted in equation of the circle its eq. will be x² + y² = a². it is 
called auxiliary circle. For auxiliary circle, diameter is equal to 
length of major axis and e = 0 
? Latus rectum:Chord through foci perpendicular to major axis 
called latus rectum. 
? Hyperbola:It is the set of all points in a plane, the differences of 
whose distance from two fixed points in the plane is a constant. 
 
  
Main facts about the Hyperbola 
 
Equation   
2 2
2 2
1
x y
a b
? ?
   
2 2
2 2
1
x y
b a
? ?
 
     a > 0, b > 0   a > 0, b > 0  
 
 
 Latus Rectum:Chord through foci perpendicular to transverse axis is 
called latus rectum. 
If 2 e ? for hyperbola, then hyperbola is called rectangular hyperbola. 
For 2 e ? then b = a and eq. of its hyperbola will be 
2 2 2
x y a ? ? or 
2 2 2
y x a ? ? . 
(A) Circle:If the equation of the circle is in the form of 
2 2
2 2 0 x y gx fy c ? ? ? ? ? 
   
then centre of circle of = c (–g, –f) 
and radius of circle =
2 2
r g f c ? ? ? 
(B) Parabola:in figure:- 
(i) Equation of the parabola y² = 4ax.