Detailed Notes: Ellipse | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


 
 
 
 
188 Conic Section : Ellipse  
   
 
 
 
 
 
 5.2.1 Definition. 
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). 
In other words, we can say an ellipse is the locus of a point which moves in a plane so that the sum of its distances 
from two fixed points is constant and is more than the distance between the two fixed points. 
Let ) , ( ? ? S is the focus, Z Z ? is the directrix and P is any point on the ellipse. Then by definition, 
PM e SP e
PM
SP
. ? ? ? 
2 2
2 2
) ( ) (
B A
C By Ax
e y x
?
? ?
? ? ? ? ? ? 
Squaring both sides, 
2 2 2 2 2 2
) ( ] ) ( ) )[( ( C By Ax e y x B A ? ? ? ? ? ? ? ? ? 
 Note  : ? The condition for second degree equation in x and y to represent an ellipse is that 0
2
? ? ab h and 
0 2
2 2 2
? ? ? ? ? ? ? ch bg af fgh abc 
 
  Example: 1 The equation of an ellipse whose focus is (–1, 1), whose directrix is 0 3 ? ? ? y x and whose eccentricity is ,
2
1
is given by 
[MP PET 1993] 
 (a) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x (b) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x  
 (c) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x (d) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x 
Solution: (a) Let any point on it be ) , ( y x then by definition, 
 
2 2
2 2
1 1
3
2
1
) 1 ( ) 1 (
?
? ?
? ? ? ?
y x
y x 
 Squaring and simplifying, we get  
 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x , which is the required ellipse. 
 5.2.2 Standard equation of the Ellipse . 
Let S be the focus, ZM be the directrix of the ellipse and ) , ( y x P is any point on the ellipse, then by definition 
e
PM
SP
? ? 
2 2 2
) ( ) ( PM e SP ? 
2
2 2 2
) 0 ( ) ( ?
?
?
?
?
?
? ? ? ? ? x
e
a
e y ae x ? 1
) 1 (
2 2
2
2
?
?
?
?
e a
y
a
x
 
1
2
2
2
2
? ?
b
y
a
x
, where ) 1 (
2 2 2
e a b ? ? 
A ? 
(–a,0) 
(0,–b) B ? 
Y ? 
x=–a/e x=a/e 
A 
(a,0) 
Z ? 
M 
(0,b) 
C 
S(ae,0) S ?(–ae,0) 
Directrix 
Directrix 
Y 
M ? 
Z 
p(x,y) 
Axis 
X 
X ? 
P(x,y) 
S( ?, ?) Ax+By+C=0 
Directrix 
Z ? 
Focus 
Z 
Page 2


 
 
 
 
188 Conic Section : Ellipse  
   
 
 
 
 
 
 5.2.1 Definition. 
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). 
In other words, we can say an ellipse is the locus of a point which moves in a plane so that the sum of its distances 
from two fixed points is constant and is more than the distance between the two fixed points. 
Let ) , ( ? ? S is the focus, Z Z ? is the directrix and P is any point on the ellipse. Then by definition, 
PM e SP e
PM
SP
. ? ? ? 
2 2
2 2
) ( ) (
B A
C By Ax
e y x
?
? ?
? ? ? ? ? ? 
Squaring both sides, 
2 2 2 2 2 2
) ( ] ) ( ) )[( ( C By Ax e y x B A ? ? ? ? ? ? ? ? ? 
 Note  : ? The condition for second degree equation in x and y to represent an ellipse is that 0
2
? ? ab h and 
0 2
2 2 2
? ? ? ? ? ? ? ch bg af fgh abc 
 
  Example: 1 The equation of an ellipse whose focus is (–1, 1), whose directrix is 0 3 ? ? ? y x and whose eccentricity is ,
2
1
is given by 
[MP PET 1993] 
 (a) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x (b) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x  
 (c) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x (d) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x 
Solution: (a) Let any point on it be ) , ( y x then by definition, 
 
2 2
2 2
1 1
3
2
1
) 1 ( ) 1 (
?
? ?
? ? ? ?
y x
y x 
 Squaring and simplifying, we get  
 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x , which is the required ellipse. 
 5.2.2 Standard equation of the Ellipse . 
Let S be the focus, ZM be the directrix of the ellipse and ) , ( y x P is any point on the ellipse, then by definition 
e
PM
SP
? ? 
2 2 2
) ( ) ( PM e SP ? 
2
2 2 2
) 0 ( ) ( ?
?
?
?
?
?
? ? ? ? ? x
e
a
e y ae x ? 1
) 1 (
2 2
2
2
?
?
?
?
e a
y
a
x
 
1
2
2
2
2
? ?
b
y
a
x
, where ) 1 (
2 2 2
e a b ? ? 
A ? 
(–a,0) 
(0,–b) B ? 
Y ? 
x=–a/e x=a/e 
A 
(a,0) 
Z ? 
M 
(0,b) 
C 
S(ae,0) S ?(–ae,0) 
Directrix 
Directrix 
Y 
M ? 
Z 
p(x,y) 
Axis 
X 
X ? 
P(x,y) 
S( ?, ?) Ax+By+C=0 
Directrix 
Z ? 
Focus 
Z 
Conic Section : Ellipse 189 
Since 1 ? e , therefore 
2 2 2
) 1 ( a e a ? ? ? 
2 2
a b ? . Some terms related to the ellipse b a
b
y
a
x
? ? ? , 1
2
2
2
2
: 
(1) Centre: The point which bisects each chord of the ellipse passing through it, is called centre ) 0 , 0 ( denoted by 
C.  
 
 
 
 
 
(2) Major and minor axes: The diameter through the foci, is called the major axis and the diameter bisecting it at 
right angles is called the minor axis. The major and minor axes are together called principal axes.  
Length of the major axis a A A 2 ? ? , Length of the minor axis b BB 2 ' ? 
The ellipse , 1
2
2
2
2
? ?
b
y
a
x
is symmetrical about both the axes. 
(3) Vertices: The extremities of the major axis of an ellipse are called vertices. 
The coordinates of vertices A and A ? are (a, 0) and (–a, 0) respectively. 
(4) Foci: S and S ? are two foci of the ellipse and their coordinates are (ae, 0) and (–ae, 0) respectively. Distance 
between foci ae S S 2 ? ? . 
(5) Directrices: ZM and M Z ? ? are two directrices of the ellipse and their equations are 
e
a
x ? and 
e
a
x ? ? 
respectively. Distance between directrices 
e
a
Z Z
2
? ? . 
(6) Eccentricity of the ellipse: For the ellipse 1
2
2
2
2
? ?
b
y
a
x
,  
we have 
2 2 2
) 1 ( e a b ? ? ? 
2
2
2
2
2
2
2
2
1
4
4
1 1 ?
?
?
?
?
?
? ? ? ? ? ?
a
b
a
b
a
b
e ;  
2
axis Major
axis Minor
1
?
?
?
?
?
?
?
?
? ? e 
This formula gives the eccentricity of the ellipse.   
(7) Ordinate and double ordinate: Let P be a point on the ellipse and let PN be perpendicular to the major axis 
AA’ such that PN produced meets the ellipse at P ? . Then PN is called the ordinate of P and P PN ? the double ordinate of P.  
If abscissa of P is h, then ordinate of P, 
2
2
2
2
1
a
h
b
y
? ? ? ) (
2 ?
? ? h a
a
b
y (For first quadrant) 
And ordinate of P ? is ) (
2 2
h a
a
b
y ?
?
?      (For fourth quadrant) 
Hence coordinates of P and P ? are ?
?
?
?
?
?
? ) ( ,
2 2
h a
a
b
h and ?
?
?
?
?
?
?
?
) ( ,
2 2
h a
a
b
h respectively. 
(8) Latus-rectum: Chord through the focus and perpendicular to the major axis is called its latus rectum. 
The double ordinates L L ?
and 
1 1
L L ? are latus rectum of the ellipse. 
Length of latus rectum 
a
b
L L LL
2
1 1
2
' ? ? ? and end points of latus-rectum are 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
a
b
ae L
a
b
ae L
2 2
, ' , , and 
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
? ?
a
b
ae L
a
b
ae L
2
1
2
1
, ' ; , 
(9) Focal chord: A chord of the ellipse passing through its focus is called a focal chord. 
A ? 
B ? 
A Z ? 
M 
C 
Y 
M ? 
Z 
X 
L ? 1 
L ? 
N S 
S ? 
L 1 L 
B P 
P ? 
Y
? 
X' 
Page 3


 
 
 
 
188 Conic Section : Ellipse  
   
 
 
 
 
 
 5.2.1 Definition. 
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). 
In other words, we can say an ellipse is the locus of a point which moves in a plane so that the sum of its distances 
from two fixed points is constant and is more than the distance between the two fixed points. 
Let ) , ( ? ? S is the focus, Z Z ? is the directrix and P is any point on the ellipse. Then by definition, 
PM e SP e
PM
SP
. ? ? ? 
2 2
2 2
) ( ) (
B A
C By Ax
e y x
?
? ?
? ? ? ? ? ? 
Squaring both sides, 
2 2 2 2 2 2
) ( ] ) ( ) )[( ( C By Ax e y x B A ? ? ? ? ? ? ? ? ? 
 Note  : ? The condition for second degree equation in x and y to represent an ellipse is that 0
2
? ? ab h and 
0 2
2 2 2
? ? ? ? ? ? ? ch bg af fgh abc 
 
  Example: 1 The equation of an ellipse whose focus is (–1, 1), whose directrix is 0 3 ? ? ? y x and whose eccentricity is ,
2
1
is given by 
[MP PET 1993] 
 (a) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x (b) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x  
 (c) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x (d) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x 
Solution: (a) Let any point on it be ) , ( y x then by definition, 
 
2 2
2 2
1 1
3
2
1
) 1 ( ) 1 (
?
? ?
? ? ? ?
y x
y x 
 Squaring and simplifying, we get  
 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x , which is the required ellipse. 
 5.2.2 Standard equation of the Ellipse . 
Let S be the focus, ZM be the directrix of the ellipse and ) , ( y x P is any point on the ellipse, then by definition 
e
PM
SP
? ? 
2 2 2
) ( ) ( PM e SP ? 
2
2 2 2
) 0 ( ) ( ?
?
?
?
?
?
? ? ? ? ? x
e
a
e y ae x ? 1
) 1 (
2 2
2
2
?
?
?
?
e a
y
a
x
 
1
2
2
2
2
? ?
b
y
a
x
, where ) 1 (
2 2 2
e a b ? ? 
A ? 
(–a,0) 
(0,–b) B ? 
Y ? 
x=–a/e x=a/e 
A 
(a,0) 
Z ? 
M 
(0,b) 
C 
S(ae,0) S ?(–ae,0) 
Directrix 
Directrix 
Y 
M ? 
Z 
p(x,y) 
Axis 
X 
X ? 
P(x,y) 
S( ?, ?) Ax+By+C=0 
Directrix 
Z ? 
Focus 
Z 
Conic Section : Ellipse 189 
Since 1 ? e , therefore 
2 2 2
) 1 ( a e a ? ? ? 
2 2
a b ? . Some terms related to the ellipse b a
b
y
a
x
? ? ? , 1
2
2
2
2
: 
(1) Centre: The point which bisects each chord of the ellipse passing through it, is called centre ) 0 , 0 ( denoted by 
C.  
 
 
 
 
 
(2) Major and minor axes: The diameter through the foci, is called the major axis and the diameter bisecting it at 
right angles is called the minor axis. The major and minor axes are together called principal axes.  
Length of the major axis a A A 2 ? ? , Length of the minor axis b BB 2 ' ? 
The ellipse , 1
2
2
2
2
? ?
b
y
a
x
is symmetrical about both the axes. 
(3) Vertices: The extremities of the major axis of an ellipse are called vertices. 
The coordinates of vertices A and A ? are (a, 0) and (–a, 0) respectively. 
(4) Foci: S and S ? are two foci of the ellipse and their coordinates are (ae, 0) and (–ae, 0) respectively. Distance 
between foci ae S S 2 ? ? . 
(5) Directrices: ZM and M Z ? ? are two directrices of the ellipse and their equations are 
e
a
x ? and 
e
a
x ? ? 
respectively. Distance between directrices 
e
a
Z Z
2
? ? . 
(6) Eccentricity of the ellipse: For the ellipse 1
2
2
2
2
? ?
b
y
a
x
,  
we have 
2 2 2
) 1 ( e a b ? ? ? 
2
2
2
2
2
2
2
2
1
4
4
1 1 ?
?
?
?
?
?
? ? ? ? ? ?
a
b
a
b
a
b
e ;  
2
axis Major
axis Minor
1
?
?
?
?
?
?
?
?
? ? e 
This formula gives the eccentricity of the ellipse.   
(7) Ordinate and double ordinate: Let P be a point on the ellipse and let PN be perpendicular to the major axis 
AA’ such that PN produced meets the ellipse at P ? . Then PN is called the ordinate of P and P PN ? the double ordinate of P.  
If abscissa of P is h, then ordinate of P, 
2
2
2
2
1
a
h
b
y
? ? ? ) (
2 ?
? ? h a
a
b
y (For first quadrant) 
And ordinate of P ? is ) (
2 2
h a
a
b
y ?
?
?      (For fourth quadrant) 
Hence coordinates of P and P ? are ?
?
?
?
?
?
? ) ( ,
2 2
h a
a
b
h and ?
?
?
?
?
?
?
?
) ( ,
2 2
h a
a
b
h respectively. 
(8) Latus-rectum: Chord through the focus and perpendicular to the major axis is called its latus rectum. 
The double ordinates L L ?
and 
1 1
L L ? are latus rectum of the ellipse. 
Length of latus rectum 
a
b
L L LL
2
1 1
2
' ? ? ? and end points of latus-rectum are 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
a
b
ae L
a
b
ae L
2 2
, ' , , and 
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
? ?
a
b
ae L
a
b
ae L
2
1
2
1
, ' ; , 
(9) Focal chord: A chord of the ellipse passing through its focus is called a focal chord. 
A ? 
B ? 
A Z ? 
M 
C 
Y 
M ? 
Z 
X 
L ? 1 
L ? 
N S 
S ? 
L 1 L 
B P 
P ? 
Y
? 
X' 
 
 
 
 
190 Conic Section : Ellipse  
(10) Focal distances of a point: The distance of a point from the focus is its focal distance. The sum of the 
focal distances of any point on an ellipse is constant and equal to the length of the 
major axis of the ellipse.  
Let ) , (
1 1
y x P be any point on the ellipse 1
2
2
2
2
? ?
b
y
a
x
 
1 1
ex a x
e
a
e ePM SP ? ? ?
?
?
?
?
?
? ? ? and 
1 1
' ' ex a x
e
a
e ePM P S ? ? ?
?
?
?
?
?
? ? ?  
?    ? ? ? ? ? ? ? ? ' 2 ) ( ) ( '
1 1
AA a ex a ex a P S SP major axis.  
 
Example: 2 The length of the latus-rectum of the ellipse 45 9 5
2 2
? ? y x is                       [MNR 1978, 80, 81; Kurukshetra CEE 1999] 
 (a) 4 / 5 (b) 2 / 5 (c) 3 / 5 (d) 3 / 10 
Solution: (d) Here the ellipse is 1
5 9
2 2
? ?
y x
 
 Here 9
2
? a and 5
2
? b . So, latus-rectum 
a
b
2
2
? =
3
10
3
) 5 ( 2
? . 
Example: 3    In an ellipse the distance between its foci is 6 and its minor axis is 8. Then its eccentricity is    [EAMCET 1994] 
 (a) 
5
4
 (b) 
52
1
 (c) 
5
3
 (d) 
2
1
 
Solution: (c) Distance between foci 6 ? ? 6 2 ? ae ? 3 ? ae , Minor axis ? ? 8 8 2 ? b ? 4 ? b ? 16
2
? b 
 From ), 1 (
2 2
e a b ? ?
?
 ? 
2 2 2
16 e a a ? ? ? 16 9
2
? ? a ? 5 ? a 
 Hence, 3 ? ae ? 
5
3
? e  
Example: 4  What is the equation of the ellipse with foci ) 0 , 2 ( ? and eccentricity 
2
1
                 [DCE 1999] 
 (a) 48 4 3
2 2
? ? y x (b) 48 3 4
2 2
? ? y x (c) 0 4 3
2 2
? ? y x (d) 0 3 4
2 2
? ? y x 
Solution: (a) Here , 2 ? ? ae 4 ,
2
1
? ? ? ? a e ?  
 Form ) 1 (
2 2 2
e a b ? ? ? ?
?
?
?
?
?
? ?
4
1
1 16
2
b ? 12
2
? b 
 Hence, the equation of ellipse is 1
12 16
2 2
? ?
y x
 or 48 4 3
2 2
? ? y x 
Example: 5     If ) 0 , 3 ( ), , (
1
? F y x P , ) 0 , 3 (
2
? ? F and 400 25 16
2 2
? ? y x , then 
2 1
PF PF ? equals                [IIT 1998] 
 (a) 8 (b) 6 (c) 10 (d) 12 
Solution: (c) We have 1
16 25
400 25 16
2 2
2 2
? ? ? ? ?
y x
y x or 1
2
2
2
2
? ?
b
y
a
x
, where 25
2
? a and 16
2
? b 
 This equation represents an ellipse with eccentricity given by  
25
9
25
16
1 1
2
2
2
? ? ? ? ?
a
b
e ? 5 / 3 ? e 
 So, the coordinates of the foci are ) 0 , ( ae ? i.e. ) 0 , 3 ( and ) 0 , 3 ( ? , Thus, 
1
F and 
2
F are the foci of the ellipse. 
 Since, the sum of the focal distance of a point on an ellipse is equal to its major axis, ? 10 2
2 1
? ? ? a PF PF   
Example: 6 An ellipse has OB as semi minor axis. F and F ? are its foci and the angle F FB ? is a right angle. Then the eccentricity of the 
ellipse is                                        [Pb. CET 2001, IIT 1997, DCE  2002] 
 (a) 
2
1
 (b) 
2
1
 (c) 
3
2
 (d) 
3
1
 
A ? 
B ? 
A Z ? 
M 
C 
Y 
M ? 
Z 
X 
S S ? 
P B 
X ? 
Y ? 
Page 4


 
 
 
 
188 Conic Section : Ellipse  
   
 
 
 
 
 
 5.2.1 Definition. 
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). 
In other words, we can say an ellipse is the locus of a point which moves in a plane so that the sum of its distances 
from two fixed points is constant and is more than the distance between the two fixed points. 
Let ) , ( ? ? S is the focus, Z Z ? is the directrix and P is any point on the ellipse. Then by definition, 
PM e SP e
PM
SP
. ? ? ? 
2 2
2 2
) ( ) (
B A
C By Ax
e y x
?
? ?
? ? ? ? ? ? 
Squaring both sides, 
2 2 2 2 2 2
) ( ] ) ( ) )[( ( C By Ax e y x B A ? ? ? ? ? ? ? ? ? 
 Note  : ? The condition for second degree equation in x and y to represent an ellipse is that 0
2
? ? ab h and 
0 2
2 2 2
? ? ? ? ? ? ? ch bg af fgh abc 
 
  Example: 1 The equation of an ellipse whose focus is (–1, 1), whose directrix is 0 3 ? ? ? y x and whose eccentricity is ,
2
1
is given by 
[MP PET 1993] 
 (a) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x (b) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x  
 (c) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x (d) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x 
Solution: (a) Let any point on it be ) , ( y x then by definition, 
 
2 2
2 2
1 1
3
2
1
) 1 ( ) 1 (
?
? ?
? ? ? ?
y x
y x 
 Squaring and simplifying, we get  
 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x , which is the required ellipse. 
 5.2.2 Standard equation of the Ellipse . 
Let S be the focus, ZM be the directrix of the ellipse and ) , ( y x P is any point on the ellipse, then by definition 
e
PM
SP
? ? 
2 2 2
) ( ) ( PM e SP ? 
2
2 2 2
) 0 ( ) ( ?
?
?
?
?
?
? ? ? ? ? x
e
a
e y ae x ? 1
) 1 (
2 2
2
2
?
?
?
?
e a
y
a
x
 
1
2
2
2
2
? ?
b
y
a
x
, where ) 1 (
2 2 2
e a b ? ? 
A ? 
(–a,0) 
(0,–b) B ? 
Y ? 
x=–a/e x=a/e 
A 
(a,0) 
Z ? 
M 
(0,b) 
C 
S(ae,0) S ?(–ae,0) 
Directrix 
Directrix 
Y 
M ? 
Z 
p(x,y) 
Axis 
X 
X ? 
P(x,y) 
S( ?, ?) Ax+By+C=0 
Directrix 
Z ? 
Focus 
Z 
Conic Section : Ellipse 189 
Since 1 ? e , therefore 
2 2 2
) 1 ( a e a ? ? ? 
2 2
a b ? . Some terms related to the ellipse b a
b
y
a
x
? ? ? , 1
2
2
2
2
: 
(1) Centre: The point which bisects each chord of the ellipse passing through it, is called centre ) 0 , 0 ( denoted by 
C.  
 
 
 
 
 
(2) Major and minor axes: The diameter through the foci, is called the major axis and the diameter bisecting it at 
right angles is called the minor axis. The major and minor axes are together called principal axes.  
Length of the major axis a A A 2 ? ? , Length of the minor axis b BB 2 ' ? 
The ellipse , 1
2
2
2
2
? ?
b
y
a
x
is symmetrical about both the axes. 
(3) Vertices: The extremities of the major axis of an ellipse are called vertices. 
The coordinates of vertices A and A ? are (a, 0) and (–a, 0) respectively. 
(4) Foci: S and S ? are two foci of the ellipse and their coordinates are (ae, 0) and (–ae, 0) respectively. Distance 
between foci ae S S 2 ? ? . 
(5) Directrices: ZM and M Z ? ? are two directrices of the ellipse and their equations are 
e
a
x ? and 
e
a
x ? ? 
respectively. Distance between directrices 
e
a
Z Z
2
? ? . 
(6) Eccentricity of the ellipse: For the ellipse 1
2
2
2
2
? ?
b
y
a
x
,  
we have 
2 2 2
) 1 ( e a b ? ? ? 
2
2
2
2
2
2
2
2
1
4
4
1 1 ?
?
?
?
?
?
? ? ? ? ? ?
a
b
a
b
a
b
e ;  
2
axis Major
axis Minor
1
?
?
?
?
?
?
?
?
? ? e 
This formula gives the eccentricity of the ellipse.   
(7) Ordinate and double ordinate: Let P be a point on the ellipse and let PN be perpendicular to the major axis 
AA’ such that PN produced meets the ellipse at P ? . Then PN is called the ordinate of P and P PN ? the double ordinate of P.  
If abscissa of P is h, then ordinate of P, 
2
2
2
2
1
a
h
b
y
? ? ? ) (
2 ?
? ? h a
a
b
y (For first quadrant) 
And ordinate of P ? is ) (
2 2
h a
a
b
y ?
?
?      (For fourth quadrant) 
Hence coordinates of P and P ? are ?
?
?
?
?
?
? ) ( ,
2 2
h a
a
b
h and ?
?
?
?
?
?
?
?
) ( ,
2 2
h a
a
b
h respectively. 
(8) Latus-rectum: Chord through the focus and perpendicular to the major axis is called its latus rectum. 
The double ordinates L L ?
and 
1 1
L L ? are latus rectum of the ellipse. 
Length of latus rectum 
a
b
L L LL
2
1 1
2
' ? ? ? and end points of latus-rectum are 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
a
b
ae L
a
b
ae L
2 2
, ' , , and 
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
? ?
a
b
ae L
a
b
ae L
2
1
2
1
, ' ; , 
(9) Focal chord: A chord of the ellipse passing through its focus is called a focal chord. 
A ? 
B ? 
A Z ? 
M 
C 
Y 
M ? 
Z 
X 
L ? 1 
L ? 
N S 
S ? 
L 1 L 
B P 
P ? 
Y
? 
X' 
 
 
 
 
190 Conic Section : Ellipse  
(10) Focal distances of a point: The distance of a point from the focus is its focal distance. The sum of the 
focal distances of any point on an ellipse is constant and equal to the length of the 
major axis of the ellipse.  
Let ) , (
1 1
y x P be any point on the ellipse 1
2
2
2
2
? ?
b
y
a
x
 
1 1
ex a x
e
a
e ePM SP ? ? ?
?
?
?
?
?
? ? ? and 
1 1
' ' ex a x
e
a
e ePM P S ? ? ?
?
?
?
?
?
? ? ?  
?    ? ? ? ? ? ? ? ? ' 2 ) ( ) ( '
1 1
AA a ex a ex a P S SP major axis.  
 
Example: 2 The length of the latus-rectum of the ellipse 45 9 5
2 2
? ? y x is                       [MNR 1978, 80, 81; Kurukshetra CEE 1999] 
 (a) 4 / 5 (b) 2 / 5 (c) 3 / 5 (d) 3 / 10 
Solution: (d) Here the ellipse is 1
5 9
2 2
? ?
y x
 
 Here 9
2
? a and 5
2
? b . So, latus-rectum 
a
b
2
2
? =
3
10
3
) 5 ( 2
? . 
Example: 3    In an ellipse the distance between its foci is 6 and its minor axis is 8. Then its eccentricity is    [EAMCET 1994] 
 (a) 
5
4
 (b) 
52
1
 (c) 
5
3
 (d) 
2
1
 
Solution: (c) Distance between foci 6 ? ? 6 2 ? ae ? 3 ? ae , Minor axis ? ? 8 8 2 ? b ? 4 ? b ? 16
2
? b 
 From ), 1 (
2 2
e a b ? ?
?
 ? 
2 2 2
16 e a a ? ? ? 16 9
2
? ? a ? 5 ? a 
 Hence, 3 ? ae ? 
5
3
? e  
Example: 4  What is the equation of the ellipse with foci ) 0 , 2 ( ? and eccentricity 
2
1
                 [DCE 1999] 
 (a) 48 4 3
2 2
? ? y x (b) 48 3 4
2 2
? ? y x (c) 0 4 3
2 2
? ? y x (d) 0 3 4
2 2
? ? y x 
Solution: (a) Here , 2 ? ? ae 4 ,
2
1
? ? ? ? a e ?  
 Form ) 1 (
2 2 2
e a b ? ? ? ?
?
?
?
?
?
? ?
4
1
1 16
2
b ? 12
2
? b 
 Hence, the equation of ellipse is 1
12 16
2 2
? ?
y x
 or 48 4 3
2 2
? ? y x 
Example: 5     If ) 0 , 3 ( ), , (
1
? F y x P , ) 0 , 3 (
2
? ? F and 400 25 16
2 2
? ? y x , then 
2 1
PF PF ? equals                [IIT 1998] 
 (a) 8 (b) 6 (c) 10 (d) 12 
Solution: (c) We have 1
16 25
400 25 16
2 2
2 2
? ? ? ? ?
y x
y x or 1
2
2
2
2
? ?
b
y
a
x
, where 25
2
? a and 16
2
? b 
 This equation represents an ellipse with eccentricity given by  
25
9
25
16
1 1
2
2
2
? ? ? ? ?
a
b
e ? 5 / 3 ? e 
 So, the coordinates of the foci are ) 0 , ( ae ? i.e. ) 0 , 3 ( and ) 0 , 3 ( ? , Thus, 
1
F and 
2
F are the foci of the ellipse. 
 Since, the sum of the focal distance of a point on an ellipse is equal to its major axis, ? 10 2
2 1
? ? ? a PF PF   
Example: 6 An ellipse has OB as semi minor axis. F and F ? are its foci and the angle F FB ? is a right angle. Then the eccentricity of the 
ellipse is                                        [Pb. CET 2001, IIT 1997, DCE  2002] 
 (a) 
2
1
 (b) 
2
1
 (c) 
3
2
 (d) 
3
1
 
A ? 
B ? 
A Z ? 
M 
C 
Y 
M ? 
Z 
X 
S S ? 
P B 
X ? 
Y ? 
 
 
 
  
Conic Section : Ellipse 191 
Solution: (b) Since 
2
'
?
? ?FBF 
 ?    
4
'
?
? ? ? ? BC F FBC 
  ?   ae b CF CB ? ? ? 
 
?  
2 2 2
e a b ?  ? 
2 2 2 2
) 1 ( e a e a ? ?  
? 
2 2
1 e e ? ? ? 1 2
2
? e ? 
2
1
? e . 
Example: 7     Let P be a variable point on the ellipse 1
2
2
2
2
? ?
b
y
a
x
with foci 
1
F and 
2
F . If A is the area of the triangle ,
2 1
F PF then the 
maximum value of A is                      [IIT 1994] 
 (a) abe 2 (b) abe (c) abe
2
1
 (d) None of these 
Solution: (b) Let ) sin , cos ( ? ? b a P and ) 0 , ( ), 0 , (
2 1
ae F ae F ? 
 ? A Area of 
2 1
F PF ? 
1 0
1 0
1 sin cos
2
1
ae
ae
b a
?
?
? ?
| sin 2 |
2
1
? aeb ? | sin | ? aeb ? 
 ?     A is maximum, when 1 | sin | ? ? .   
 Hence, maximum value of abe A ? 
 Example: 8  The eccentricity of an ellipse, with its centre at the origin is 
2
1
. If one of the directrices is 4 ? x , then the equation of the ellipse 
is                [AIEEE 2004] 
 (a) 1 3 4
2 2
? ? y x (b) 12 4 3
2 2
? ? y x (c) 12 3 4
2 2
? ? y x (d) 1 4 3
2 2
? ? y x 
Solution: (b) Given 4 ,
2
1
? ?
e
a
e . So, 4 2
2
? ? ? a a 
 From ) 1 (
2 2 2
e a b ? ? ?  3
4
3
4
4
1
1 4
2
? ? ? ?
?
?
?
?
?
? ? b 
 Hence the equation of ellipse is 1
3 4
2 2
? ?
y x
,  i.e. 12 4 3
2 2
? ? y x    
 5.2.3 Equation of Ellipse in other form . 
In the equation of the ellipse , 1
2
2
2
2
? ?
b
y
a
x
if b a ? or 
2 2
b a ? (denominator of 
2
x is 
greater than that of 
2
y ), then the major and minor axis lie along x-axis and          y-axis 
respectively. But if b a ? or 
2 2
b a ? (denominator of 
2
x is less than that of 
2
y ), then the 
major axis of the ellipse lies along the y-axis and is of length 2b and the minor axis along the x-
axis and is of length 2a.  
The coordinates of foci S and S’ are (0, be) and (0, – be) respectively.  
The equation of the directrices ZK and ' ' K Z are e b y / ? ? and eccentricity e is given 
by the formula ) 1 (
2 2 2
e b a ? ? or 
2
2
1
b
a
e ? ? 
 
B 
Y 
 
 
 
X 
C 
X’ 
Y ? 
 
 
 
F ? F 
S ? 
(0,0) 
(0,be) 
(0,–be) 
B ? 
(–a,0) 
B 
(a,0) 
C 
A(0,b) 
Z 
Y 
K y=b/e 
y=–b/e 
Y ? 
K ? 
A ?(0,– b) 
X X ? 
Z ? 
Page 5


 
 
 
 
188 Conic Section : Ellipse  
   
 
 
 
 
 
 5.2.1 Definition. 
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). 
In other words, we can say an ellipse is the locus of a point which moves in a plane so that the sum of its distances 
from two fixed points is constant and is more than the distance between the two fixed points. 
Let ) , ( ? ? S is the focus, Z Z ? is the directrix and P is any point on the ellipse. Then by definition, 
PM e SP e
PM
SP
. ? ? ? 
2 2
2 2
) ( ) (
B A
C By Ax
e y x
?
? ?
? ? ? ? ? ? 
Squaring both sides, 
2 2 2 2 2 2
) ( ] ) ( ) )[( ( C By Ax e y x B A ? ? ? ? ? ? ? ? ? 
 Note  : ? The condition for second degree equation in x and y to represent an ellipse is that 0
2
? ? ab h and 
0 2
2 2 2
? ? ? ? ? ? ? ch bg af fgh abc 
 
  Example: 1 The equation of an ellipse whose focus is (–1, 1), whose directrix is 0 3 ? ? ? y x and whose eccentricity is ,
2
1
is given by 
[MP PET 1993] 
 (a) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x (b) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x  
 (c) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x (d) 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x 
Solution: (a) Let any point on it be ) , ( y x then by definition, 
 
2 2
2 2
1 1
3
2
1
) 1 ( ) 1 (
?
? ?
? ? ? ?
y x
y x 
 Squaring and simplifying, we get  
 0 7 10 10 7 2 7
2 2
? ? ? ? ? ? y x y xy x , which is the required ellipse. 
 5.2.2 Standard equation of the Ellipse . 
Let S be the focus, ZM be the directrix of the ellipse and ) , ( y x P is any point on the ellipse, then by definition 
e
PM
SP
? ? 
2 2 2
) ( ) ( PM e SP ? 
2
2 2 2
) 0 ( ) ( ?
?
?
?
?
?
? ? ? ? ? x
e
a
e y ae x ? 1
) 1 (
2 2
2
2
?
?
?
?
e a
y
a
x
 
1
2
2
2
2
? ?
b
y
a
x
, where ) 1 (
2 2 2
e a b ? ? 
A ? 
(–a,0) 
(0,–b) B ? 
Y ? 
x=–a/e x=a/e 
A 
(a,0) 
Z ? 
M 
(0,b) 
C 
S(ae,0) S ?(–ae,0) 
Directrix 
Directrix 
Y 
M ? 
Z 
p(x,y) 
Axis 
X 
X ? 
P(x,y) 
S( ?, ?) Ax+By+C=0 
Directrix 
Z ? 
Focus 
Z 
Conic Section : Ellipse 189 
Since 1 ? e , therefore 
2 2 2
) 1 ( a e a ? ? ? 
2 2
a b ? . Some terms related to the ellipse b a
b
y
a
x
? ? ? , 1
2
2
2
2
: 
(1) Centre: The point which bisects each chord of the ellipse passing through it, is called centre ) 0 , 0 ( denoted by 
C.  
 
 
 
 
 
(2) Major and minor axes: The diameter through the foci, is called the major axis and the diameter bisecting it at 
right angles is called the minor axis. The major and minor axes are together called principal axes.  
Length of the major axis a A A 2 ? ? , Length of the minor axis b BB 2 ' ? 
The ellipse , 1
2
2
2
2
? ?
b
y
a
x
is symmetrical about both the axes. 
(3) Vertices: The extremities of the major axis of an ellipse are called vertices. 
The coordinates of vertices A and A ? are (a, 0) and (–a, 0) respectively. 
(4) Foci: S and S ? are two foci of the ellipse and their coordinates are (ae, 0) and (–ae, 0) respectively. Distance 
between foci ae S S 2 ? ? . 
(5) Directrices: ZM and M Z ? ? are two directrices of the ellipse and their equations are 
e
a
x ? and 
e
a
x ? ? 
respectively. Distance between directrices 
e
a
Z Z
2
? ? . 
(6) Eccentricity of the ellipse: For the ellipse 1
2
2
2
2
? ?
b
y
a
x
,  
we have 
2 2 2
) 1 ( e a b ? ? ? 
2
2
2
2
2
2
2
2
1
4
4
1 1 ?
?
?
?
?
?
? ? ? ? ? ?
a
b
a
b
a
b
e ;  
2
axis Major
axis Minor
1
?
?
?
?
?
?
?
?
? ? e 
This formula gives the eccentricity of the ellipse.   
(7) Ordinate and double ordinate: Let P be a point on the ellipse and let PN be perpendicular to the major axis 
AA’ such that PN produced meets the ellipse at P ? . Then PN is called the ordinate of P and P PN ? the double ordinate of P.  
If abscissa of P is h, then ordinate of P, 
2
2
2
2
1
a
h
b
y
? ? ? ) (
2 ?
? ? h a
a
b
y (For first quadrant) 
And ordinate of P ? is ) (
2 2
h a
a
b
y ?
?
?      (For fourth quadrant) 
Hence coordinates of P and P ? are ?
?
?
?
?
?
? ) ( ,
2 2
h a
a
b
h and ?
?
?
?
?
?
?
?
) ( ,
2 2
h a
a
b
h respectively. 
(8) Latus-rectum: Chord through the focus and perpendicular to the major axis is called its latus rectum. 
The double ordinates L L ?
and 
1 1
L L ? are latus rectum of the ellipse. 
Length of latus rectum 
a
b
L L LL
2
1 1
2
' ? ? ? and end points of latus-rectum are 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
a
b
ae L
a
b
ae L
2 2
, ' , , and 
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
? ?
a
b
ae L
a
b
ae L
2
1
2
1
, ' ; , 
(9) Focal chord: A chord of the ellipse passing through its focus is called a focal chord. 
A ? 
B ? 
A Z ? 
M 
C 
Y 
M ? 
Z 
X 
L ? 1 
L ? 
N S 
S ? 
L 1 L 
B P 
P ? 
Y
? 
X' 
 
 
 
 
190 Conic Section : Ellipse  
(10) Focal distances of a point: The distance of a point from the focus is its focal distance. The sum of the 
focal distances of any point on an ellipse is constant and equal to the length of the 
major axis of the ellipse.  
Let ) , (
1 1
y x P be any point on the ellipse 1
2
2
2
2
? ?
b
y
a
x
 
1 1
ex a x
e
a
e ePM SP ? ? ?
?
?
?
?
?
? ? ? and 
1 1
' ' ex a x
e
a
e ePM P S ? ? ?
?
?
?
?
?
? ? ?  
?    ? ? ? ? ? ? ? ? ' 2 ) ( ) ( '
1 1
AA a ex a ex a P S SP major axis.  
 
Example: 2 The length of the latus-rectum of the ellipse 45 9 5
2 2
? ? y x is                       [MNR 1978, 80, 81; Kurukshetra CEE 1999] 
 (a) 4 / 5 (b) 2 / 5 (c) 3 / 5 (d) 3 / 10 
Solution: (d) Here the ellipse is 1
5 9
2 2
? ?
y x
 
 Here 9
2
? a and 5
2
? b . So, latus-rectum 
a
b
2
2
? =
3
10
3
) 5 ( 2
? . 
Example: 3    In an ellipse the distance between its foci is 6 and its minor axis is 8. Then its eccentricity is    [EAMCET 1994] 
 (a) 
5
4
 (b) 
52
1
 (c) 
5
3
 (d) 
2
1
 
Solution: (c) Distance between foci 6 ? ? 6 2 ? ae ? 3 ? ae , Minor axis ? ? 8 8 2 ? b ? 4 ? b ? 16
2
? b 
 From ), 1 (
2 2
e a b ? ?
?
 ? 
2 2 2
16 e a a ? ? ? 16 9
2
? ? a ? 5 ? a 
 Hence, 3 ? ae ? 
5
3
? e  
Example: 4  What is the equation of the ellipse with foci ) 0 , 2 ( ? and eccentricity 
2
1
                 [DCE 1999] 
 (a) 48 4 3
2 2
? ? y x (b) 48 3 4
2 2
? ? y x (c) 0 4 3
2 2
? ? y x (d) 0 3 4
2 2
? ? y x 
Solution: (a) Here , 2 ? ? ae 4 ,
2
1
? ? ? ? a e ?  
 Form ) 1 (
2 2 2
e a b ? ? ? ?
?
?
?
?
?
? ?
4
1
1 16
2
b ? 12
2
? b 
 Hence, the equation of ellipse is 1
12 16
2 2
? ?
y x
 or 48 4 3
2 2
? ? y x 
Example: 5     If ) 0 , 3 ( ), , (
1
? F y x P , ) 0 , 3 (
2
? ? F and 400 25 16
2 2
? ? y x , then 
2 1
PF PF ? equals                [IIT 1998] 
 (a) 8 (b) 6 (c) 10 (d) 12 
Solution: (c) We have 1
16 25
400 25 16
2 2
2 2
? ? ? ? ?
y x
y x or 1
2
2
2
2
? ?
b
y
a
x
, where 25
2
? a and 16
2
? b 
 This equation represents an ellipse with eccentricity given by  
25
9
25
16
1 1
2
2
2
? ? ? ? ?
a
b
e ? 5 / 3 ? e 
 So, the coordinates of the foci are ) 0 , ( ae ? i.e. ) 0 , 3 ( and ) 0 , 3 ( ? , Thus, 
1
F and 
2
F are the foci of the ellipse. 
 Since, the sum of the focal distance of a point on an ellipse is equal to its major axis, ? 10 2
2 1
? ? ? a PF PF   
Example: 6 An ellipse has OB as semi minor axis. F and F ? are its foci and the angle F FB ? is a right angle. Then the eccentricity of the 
ellipse is                                        [Pb. CET 2001, IIT 1997, DCE  2002] 
 (a) 
2
1
 (b) 
2
1
 (c) 
3
2
 (d) 
3
1
 
A ? 
B ? 
A Z ? 
M 
C 
Y 
M ? 
Z 
X 
S S ? 
P B 
X ? 
Y ? 
 
 
 
  
Conic Section : Ellipse 191 
Solution: (b) Since 
2
'
?
? ?FBF 
 ?    
4
'
?
? ? ? ? BC F FBC 
  ?   ae b CF CB ? ? ? 
 
?  
2 2 2
e a b ?  ? 
2 2 2 2
) 1 ( e a e a ? ?  
? 
2 2
1 e e ? ? ? 1 2
2
? e ? 
2
1
? e . 
Example: 7     Let P be a variable point on the ellipse 1
2
2
2
2
? ?
b
y
a
x
with foci 
1
F and 
2
F . If A is the area of the triangle ,
2 1
F PF then the 
maximum value of A is                      [IIT 1994] 
 (a) abe 2 (b) abe (c) abe
2
1
 (d) None of these 
Solution: (b) Let ) sin , cos ( ? ? b a P and ) 0 , ( ), 0 , (
2 1
ae F ae F ? 
 ? A Area of 
2 1
F PF ? 
1 0
1 0
1 sin cos
2
1
ae
ae
b a
?
?
? ?
| sin 2 |
2
1
? aeb ? | sin | ? aeb ? 
 ?     A is maximum, when 1 | sin | ? ? .   
 Hence, maximum value of abe A ? 
 Example: 8  The eccentricity of an ellipse, with its centre at the origin is 
2
1
. If one of the directrices is 4 ? x , then the equation of the ellipse 
is                [AIEEE 2004] 
 (a) 1 3 4
2 2
? ? y x (b) 12 4 3
2 2
? ? y x (c) 12 3 4
2 2
? ? y x (d) 1 4 3
2 2
? ? y x 
Solution: (b) Given 4 ,
2
1
? ?
e
a
e . So, 4 2
2
? ? ? a a 
 From ) 1 (
2 2 2
e a b ? ? ?  3
4
3
4
4
1
1 4
2
? ? ? ?
?
?
?
?
?
? ? b 
 Hence the equation of ellipse is 1
3 4
2 2
? ?
y x
,  i.e. 12 4 3
2 2
? ? y x    
 5.2.3 Equation of Ellipse in other form . 
In the equation of the ellipse , 1
2
2
2
2
? ?
b
y
a
x
if b a ? or 
2 2
b a ? (denominator of 
2
x is 
greater than that of 
2
y ), then the major and minor axis lie along x-axis and          y-axis 
respectively. But if b a ? or 
2 2
b a ? (denominator of 
2
x is less than that of 
2
y ), then the 
major axis of the ellipse lies along the y-axis and is of length 2b and the minor axis along the x-
axis and is of length 2a.  
The coordinates of foci S and S’ are (0, be) and (0, – be) respectively.  
The equation of the directrices ZK and ' ' K Z are e b y / ? ? and eccentricity e is given 
by the formula ) 1 (
2 2 2
e b a ? ? or 
2
2
1
b
a
e ? ? 
 
B 
Y 
 
 
 
X 
C 
X’ 
Y ? 
 
 
 
F ? F 
S ? 
(0,0) 
(0,be) 
(0,–be) 
B ? 
(–a,0) 
B 
(a,0) 
C 
A(0,b) 
Z 
Y 
K y=b/e 
y=–b/e 
Y ? 
K ? 
A ?(0,– b) 
X X ? 
Z ? 
 
 
 
 
192 Conic Section : Ellipse  
Difference between both ellipse will be clear from the following table. 
 
Ellipse  
 
   Basic fundamentals 
?
?
?
?
?
?
? ? 1
2
2
2
2
b
y
a
x
 
 
For a > b For b > a 
Centre (0, 0)  (0, 0)  
Vertices ) 0 , ( a ? ) , 0 ( b ? 
Length of major axis 2a 2b 
Length of minor axis 2b 2a  
Foci ) 0 , ( ae ? ) , 0 ( be ? 
Equation of directrices e a x / ? ? e b y / ? ? 
Relation in a, b and e 
) 1 (
2 2 2
e a b ? ? ) 1 (
2 2 2
e b a ? ? 
Length of latus rectum  
a
b
2
2
 
b
a
2
2
 
Ends of latus-rectum 
?
?
?
?
?
?
?
?
? ?
a
b
ae
2
, 
?
?
?
?
?
?
?
?
? ? be
b
a
,
2
 
Parametric equations  ) sin , cos ( ? ? b a ) sin , cos ( ? ? b a ) 2 0 ( ? ? ? ? 
Focal radii 
1
ex a SP ? ? and
1
' ex a P S ? ? 
1
ey b SP ? ? and 
1
' ey b P S ? ? 
Sum of focal radii ? ? P S SP ' 2a 2b 
Distance between foci 2ae 2be 
Distance between directrices 2a/e 2b/e 
Tangents at the vertices  x = –a, x = a  y = b, y = –b 
 
Example: 9 The equation of a directrix of the ellipse 1
25 16
2 2
? ?
y x
is       
 (a) 
3
25
? y (b) 3 ? x (c) 3 ? ? x (d) 
25
3
? x 
Solution: (a) From the given equation of ellipse 25 , 16
2 2
? ? b a (since a b ? ) 
 So, ) 1 (
2 2 2
e b a ? ? , ? ) 1 ( 25 16
2
e ? ? ? 
25
9
25
16
1
2 2
? ? ? ? e e ? 
5
3
? e 
 ? One directrix is 
3
25
5 / 3
5
? ? ?
e
b
y 
Example: 10 The distances from the foci of ) , (
1 1
y x P on the ellipse 1
25 9
2 2
? ?
y x
are     
 (a) 
1
4
5
4 y ? (b) 
1
5
4
5 x ? (c) 
1
5
4
5 y ? (d) None of these 
Solution: (c) For the given ellipse , a b ? so the two foci lie on y-axis and their coordinates are ? ? be ? , 0 ,  
 Where 3 , 5 ? ? a b . So 
5
4
25
9
1 1
2
2
? ? ? ? ?
b
a
e 
 The focal distances of a point ) , (
1 1
y x on the ellipse 1
2
2
2
2
? ?
b
y
a
x
, Where 
2 2
a b ? are given by 
1
ey b ? . So, Required distances 
are 
1 1
5
4
5 y ey b ? ? ? . 
 5.2.4 Parametric form of the Ellipse.