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

 Page 1


ELLIPSE 
The ellipse is a fundamental conic section with wide-ranging applications in mathematics, physics, 
and engineering. This geometric shape, defined by its unique properties, plays a crucial role in 
various fields from planetary orbits to architectural design. 
This section provides a comprehensive exploration of ellipses, starting with their standard equation 
and key definitions. We'll examine the essential components of an ellipse, including its foci, 
vertices, axes, and eccentricity. The material covers various forms of ellipse equations, methods for 
determining the position of points relative to an ellipse, and the concept of the auxiliary circle and 
eccentric angle. 
As we delve deeper, we'll investigate the parametric representation of ellipses, their interaction 
with lines, and the properties of tangents to ellipses. Through a combination of theoretical 
explanations and practical problems, this material offers a thorough grounding in the mathematics 
of ellipses, preparing the way for more advanced study and applications in related fields. 
1. STANDARD EQUATION & DEFINITION: 
Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is 
?? 2
?? 2
+
?? 2
?? 2
=
1. where ?? > ?? &?? 2
= ?? 2
( 1 - ?? 2
) ? ?? 2
- ?? 2
= ?? 2
?? 2
. where ?? = eccentricity ( 0 < ?? < 1) . 
FOCI: S = ( ???? , 0) & ?? '
= ( -???? , 0) . 
(a) Equation of directrices: 
?? =
?? ?? & ?? = -
?? ?? 
(b) Vertices: 
?? '
= ( -?? , 0) & ?? = ( ?? , 0) . 
(c) Major axis: The line segment ?? '
?? in which the foci ?? '
& ?? lie is of length 2 a & is called the major 
axis ( ?? > ?? ) of the ellipse. Point of intersection of major axis with directrix is called the 
foot of the directrix ( ?? )(±
?? ?? , 0). 
(d) Minor Axis: The y-axis intersects the ellipse in the points ?? '
= ( 0, -?? ) & ?? = ( 0, b). The line 
segment B'B of length 2 ?? ( ?? < ?? ) is called the Minor Axis of the ellipse. 
(e) Principal Axes: The major & minor axis together are called Principal Axes of the ellipse. 
(f) Centre: The point which bisects every chord of the conic drawn through it is called the centre of 
the conic. ?? = ( 0,0) the origin is the centre of the ellipse 
?? 2
?? 2
+
?? 2
?? 2
= 1. 
(g) Diameter: A chord of the conic which passes through the centre is called a diameter of the 
conic. 
Page 2


ELLIPSE 
The ellipse is a fundamental conic section with wide-ranging applications in mathematics, physics, 
and engineering. This geometric shape, defined by its unique properties, plays a crucial role in 
various fields from planetary orbits to architectural design. 
This section provides a comprehensive exploration of ellipses, starting with their standard equation 
and key definitions. We'll examine the essential components of an ellipse, including its foci, 
vertices, axes, and eccentricity. The material covers various forms of ellipse equations, methods for 
determining the position of points relative to an ellipse, and the concept of the auxiliary circle and 
eccentric angle. 
As we delve deeper, we'll investigate the parametric representation of ellipses, their interaction 
with lines, and the properties of tangents to ellipses. Through a combination of theoretical 
explanations and practical problems, this material offers a thorough grounding in the mathematics 
of ellipses, preparing the way for more advanced study and applications in related fields. 
1. STANDARD EQUATION & DEFINITION: 
Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is 
?? 2
?? 2
+
?? 2
?? 2
=
1. where ?? > ?? &?? 2
= ?? 2
( 1 - ?? 2
) ? ?? 2
- ?? 2
= ?? 2
?? 2
. where ?? = eccentricity ( 0 < ?? < 1) . 
FOCI: S = ( ???? , 0) & ?? '
= ( -???? , 0) . 
(a) Equation of directrices: 
?? =
?? ?? & ?? = -
?? ?? 
(b) Vertices: 
?? '
= ( -?? , 0) & ?? = ( ?? , 0) . 
(c) Major axis: The line segment ?? '
?? in which the foci ?? '
& ?? lie is of length 2 a & is called the major 
axis ( ?? > ?? ) of the ellipse. Point of intersection of major axis with directrix is called the 
foot of the directrix ( ?? )(±
?? ?? , 0). 
(d) Minor Axis: The y-axis intersects the ellipse in the points ?? '
= ( 0, -?? ) & ?? = ( 0, b). The line 
segment B'B of length 2 ?? ( ?? < ?? ) is called the Minor Axis of the ellipse. 
(e) Principal Axes: The major & minor axis together are called Principal Axes of the ellipse. 
(f) Centre: The point which bisects every chord of the conic drawn through it is called the centre of 
the conic. ?? = ( 0,0) the origin is the centre of the ellipse 
?? 2
?? 2
+
?? 2
?? 2
= 1. 
(g) Diameter: A chord of the conic which passes through the centre is called a diameter of the 
conic. 
(h) Focal Chord: A chord which passes through a focus is called a focal chord. 
(i) Double Ordinate: A chord perpendicular to the major axis is called a double ordinate. 
(j) Latus Rectum: The focal chord perpendicular to the major axis is called the latus rectum. 
(i) Length of latus rectum (LL') =
2 ?? 2
?? =
( ?????????? ???????? )
2
 ?????????? ???????? 
= 2?? ( 1 - ?? 2
) 
(ii) Equation of latus rectum: ?? = ± ae. 
(iii) Ends of the latus rectum are ?? (???? ,
?? 2
?? ), ?? '
(???? , -
?? 2
?? ), ?? 1
(-???? ,
?? 2
?? ) and ?? 1
(-???? , -
?? 2
?? ). 
? ???? + ?? ' ?? = 2?? = ?????????? ???????? .  
(l) Eccentricity: ?? = v1 -
?? 2
?? 2
 
Note: 
(i) The sum of the focal distances of any point on the ellipse is equal to the major Axis. Hence 
distance of focus from the extremity of a minor axis is equal to semi major axis. i.e BS = ???? . 
(ii) If the equation of the ellipse is given as 
?? 2
?? 2
+
?? 2
?? 2
= 1& nothing is mentioned, then the rule is to 
assume that ?? > ?? . 
Problem 1: If LR of an ellipse is half of its minor axis, then its eccentricity is - 
(A) 
3
2
 
(B) 
2
3
 
(C) 
v3
2
 
(D) 
v2
3
 
Solution: 
  ???? ?????????? 
2?? 2
?? = ?? ? 2 ?? = ??   ? 4?? 2
( 1 - ?? 2
) = ?? 2
 ? 1 - ?? 2
= 1/4  
Problem 2: Find the equation of the ellipse whose foci are ( 2,3) , ( -2,3) and whose semi minor axis 
is of length v5. 
Solution:   Here ?? is ( 2,3) &?? '
 is ( -2,3) and ?? = v5 ? ?? '
= 4 = 2???? ? ae = 2 but  ?? 2
=
?? 2
( 1 - ?? 2
) ? 5 = ?? 2
- 4 ? ?? = 3. 
Hence the equation to major axis is ?? = 3 
Centre of ellipse is midpoint of SS' i.e. (0, 3) 
?  Equation to ellipse is 
?? 2
?? 2
+
( ?? -3)
2
?? 2
= 1 or 
?? 2
9
+
( ?? -3)
2
5
= 1 
Page 3


ELLIPSE 
The ellipse is a fundamental conic section with wide-ranging applications in mathematics, physics, 
and engineering. This geometric shape, defined by its unique properties, plays a crucial role in 
various fields from planetary orbits to architectural design. 
This section provides a comprehensive exploration of ellipses, starting with their standard equation 
and key definitions. We'll examine the essential components of an ellipse, including its foci, 
vertices, axes, and eccentricity. The material covers various forms of ellipse equations, methods for 
determining the position of points relative to an ellipse, and the concept of the auxiliary circle and 
eccentric angle. 
As we delve deeper, we'll investigate the parametric representation of ellipses, their interaction 
with lines, and the properties of tangents to ellipses. Through a combination of theoretical 
explanations and practical problems, this material offers a thorough grounding in the mathematics 
of ellipses, preparing the way for more advanced study and applications in related fields. 
1. STANDARD EQUATION & DEFINITION: 
Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is 
?? 2
?? 2
+
?? 2
?? 2
=
1. where ?? > ?? &?? 2
= ?? 2
( 1 - ?? 2
) ? ?? 2
- ?? 2
= ?? 2
?? 2
. where ?? = eccentricity ( 0 < ?? < 1) . 
FOCI: S = ( ???? , 0) & ?? '
= ( -???? , 0) . 
(a) Equation of directrices: 
?? =
?? ?? & ?? = -
?? ?? 
(b) Vertices: 
?? '
= ( -?? , 0) & ?? = ( ?? , 0) . 
(c) Major axis: The line segment ?? '
?? in which the foci ?? '
& ?? lie is of length 2 a & is called the major 
axis ( ?? > ?? ) of the ellipse. Point of intersection of major axis with directrix is called the 
foot of the directrix ( ?? )(±
?? ?? , 0). 
(d) Minor Axis: The y-axis intersects the ellipse in the points ?? '
= ( 0, -?? ) & ?? = ( 0, b). The line 
segment B'B of length 2 ?? ( ?? < ?? ) is called the Minor Axis of the ellipse. 
(e) Principal Axes: The major & minor axis together are called Principal Axes of the ellipse. 
(f) Centre: The point which bisects every chord of the conic drawn through it is called the centre of 
the conic. ?? = ( 0,0) the origin is the centre of the ellipse 
?? 2
?? 2
+
?? 2
?? 2
= 1. 
(g) Diameter: A chord of the conic which passes through the centre is called a diameter of the 
conic. 
(h) Focal Chord: A chord which passes through a focus is called a focal chord. 
(i) Double Ordinate: A chord perpendicular to the major axis is called a double ordinate. 
(j) Latus Rectum: The focal chord perpendicular to the major axis is called the latus rectum. 
(i) Length of latus rectum (LL') =
2 ?? 2
?? =
( ?????????? ???????? )
2
 ?????????? ???????? 
= 2?? ( 1 - ?? 2
) 
(ii) Equation of latus rectum: ?? = ± ae. 
(iii) Ends of the latus rectum are ?? (???? ,
?? 2
?? ), ?? '
(???? , -
?? 2
?? ), ?? 1
(-???? ,
?? 2
?? ) and ?? 1
(-???? , -
?? 2
?? ). 
? ???? + ?? ' ?? = 2?? = ?????????? ???????? .  
(l) Eccentricity: ?? = v1 -
?? 2
?? 2
 
Note: 
(i) The sum of the focal distances of any point on the ellipse is equal to the major Axis. Hence 
distance of focus from the extremity of a minor axis is equal to semi major axis. i.e BS = ???? . 
(ii) If the equation of the ellipse is given as 
?? 2
?? 2
+
?? 2
?? 2
= 1& nothing is mentioned, then the rule is to 
assume that ?? > ?? . 
Problem 1: If LR of an ellipse is half of its minor axis, then its eccentricity is - 
(A) 
3
2
 
(B) 
2
3
 
(C) 
v3
2
 
(D) 
v2
3
 
Solution: 
  ???? ?????????? 
2?? 2
?? = ?? ? 2 ?? = ??   ? 4?? 2
( 1 - ?? 2
) = ?? 2
 ? 1 - ?? 2
= 1/4  
Problem 2: Find the equation of the ellipse whose foci are ( 2,3) , ( -2,3) and whose semi minor axis 
is of length v5. 
Solution:   Here ?? is ( 2,3) &?? '
 is ( -2,3) and ?? = v5 ? ?? '
= 4 = 2???? ? ae = 2 but  ?? 2
=
?? 2
( 1 - ?? 2
) ? 5 = ?? 2
- 4 ? ?? = 3. 
Hence the equation to major axis is ?? = 3 
Centre of ellipse is midpoint of SS' i.e. (0, 3) 
?  Equation to ellipse is 
?? 2
?? 2
+
( ?? -3)
2
?? 2
= 1 or 
?? 2
9
+
( ?? -3)
2
5
= 1 
Ans. 
Problem 3: Find the equation of the ellipse having centre at ( 1,2) , one focus at ( 6,2) and passing 
through the point ( 4,6) . 
Solution: 
With centre at ( 1,2) , the equation of the ellipse is 
( ?? -1)
2
?? 2
+
( ?? -2)
2
?? 2
= 1. It passes through the point 
( 4,6) 
? 
9
?? 2
+
16
?? 2
= 1 
Distance between the focus and the centre = ( 6 - 1) = 5 = ae 
? ?? 2
= ?? 2
- ?? 2
?? 2
= ?? 2
- 25 
Solving for ?? 2
 and ?? 2
 from the equations (i) and (ii), we get ?? 2
= 45 and ?? 2
= 20. 
Hence the equation of the ellipse is 
( ?? -1)
2
45
+
( ?? -2)
2
20
= 1 
Ans. 
Do yourself - 1: 
(i) If LR of an ellipse 
?? 2
?? 2
+
?? 2
?? 2
= 1, ( ?? < ?? ) is half of its major axis, then find its eccentricity. 
(ii) Find the equation of the ellipse whose foci are ( 4,6) &( 16,6) and whose semi-minor axis is 4 . 
(iii) Find the eccentricity, foci and the length of the latus-rectum of the ellipse ?? 2
+ 4?? 2
+ 8?? -
2?? + 1 = 0. 
2. ANOTHER FORM OF ELLIPSE: 
 
?? 2
?? 2
+
?? 2
 ?? 2
= 1, ( ?? < ?? ) 
(a) ????
'
= Minor axis = 2?? 
(b) ????
'
= Major axis = 2?? 
(c) ?? 2
= ?? 2
( 1 - ?? 2
) 
(d) Latus rectum ?? ?? '
= ?? 1
?? 1
'
=
2?? 2
?? , equation ?? = ± be 
(e) Ends of the latus rectum are: 
?? (
?? 2
 ?? , ???? ) , ?? '
(-
?? 2
 ?? , ???? ) , ?? 1
(
?? 2
 ?? , -???? ) , ?? 1
'
(-
?? 2
 ?? , -???? ) 
(f) Equation of directrix ?? = ±?? /?? 
Page 4


ELLIPSE 
The ellipse is a fundamental conic section with wide-ranging applications in mathematics, physics, 
and engineering. This geometric shape, defined by its unique properties, plays a crucial role in 
various fields from planetary orbits to architectural design. 
This section provides a comprehensive exploration of ellipses, starting with their standard equation 
and key definitions. We'll examine the essential components of an ellipse, including its foci, 
vertices, axes, and eccentricity. The material covers various forms of ellipse equations, methods for 
determining the position of points relative to an ellipse, and the concept of the auxiliary circle and 
eccentric angle. 
As we delve deeper, we'll investigate the parametric representation of ellipses, their interaction 
with lines, and the properties of tangents to ellipses. Through a combination of theoretical 
explanations and practical problems, this material offers a thorough grounding in the mathematics 
of ellipses, preparing the way for more advanced study and applications in related fields. 
1. STANDARD EQUATION & DEFINITION: 
Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is 
?? 2
?? 2
+
?? 2
?? 2
=
1. where ?? > ?? &?? 2
= ?? 2
( 1 - ?? 2
) ? ?? 2
- ?? 2
= ?? 2
?? 2
. where ?? = eccentricity ( 0 < ?? < 1) . 
FOCI: S = ( ???? , 0) & ?? '
= ( -???? , 0) . 
(a) Equation of directrices: 
?? =
?? ?? & ?? = -
?? ?? 
(b) Vertices: 
?? '
= ( -?? , 0) & ?? = ( ?? , 0) . 
(c) Major axis: The line segment ?? '
?? in which the foci ?? '
& ?? lie is of length 2 a & is called the major 
axis ( ?? > ?? ) of the ellipse. Point of intersection of major axis with directrix is called the 
foot of the directrix ( ?? )(±
?? ?? , 0). 
(d) Minor Axis: The y-axis intersects the ellipse in the points ?? '
= ( 0, -?? ) & ?? = ( 0, b). The line 
segment B'B of length 2 ?? ( ?? < ?? ) is called the Minor Axis of the ellipse. 
(e) Principal Axes: The major & minor axis together are called Principal Axes of the ellipse. 
(f) Centre: The point which bisects every chord of the conic drawn through it is called the centre of 
the conic. ?? = ( 0,0) the origin is the centre of the ellipse 
?? 2
?? 2
+
?? 2
?? 2
= 1. 
(g) Diameter: A chord of the conic which passes through the centre is called a diameter of the 
conic. 
(h) Focal Chord: A chord which passes through a focus is called a focal chord. 
(i) Double Ordinate: A chord perpendicular to the major axis is called a double ordinate. 
(j) Latus Rectum: The focal chord perpendicular to the major axis is called the latus rectum. 
(i) Length of latus rectum (LL') =
2 ?? 2
?? =
( ?????????? ???????? )
2
 ?????????? ???????? 
= 2?? ( 1 - ?? 2
) 
(ii) Equation of latus rectum: ?? = ± ae. 
(iii) Ends of the latus rectum are ?? (???? ,
?? 2
?? ), ?? '
(???? , -
?? 2
?? ), ?? 1
(-???? ,
?? 2
?? ) and ?? 1
(-???? , -
?? 2
?? ). 
? ???? + ?? ' ?? = 2?? = ?????????? ???????? .  
(l) Eccentricity: ?? = v1 -
?? 2
?? 2
 
Note: 
(i) The sum of the focal distances of any point on the ellipse is equal to the major Axis. Hence 
distance of focus from the extremity of a minor axis is equal to semi major axis. i.e BS = ???? . 
(ii) If the equation of the ellipse is given as 
?? 2
?? 2
+
?? 2
?? 2
= 1& nothing is mentioned, then the rule is to 
assume that ?? > ?? . 
Problem 1: If LR of an ellipse is half of its minor axis, then its eccentricity is - 
(A) 
3
2
 
(B) 
2
3
 
(C) 
v3
2
 
(D) 
v2
3
 
Solution: 
  ???? ?????????? 
2?? 2
?? = ?? ? 2 ?? = ??   ? 4?? 2
( 1 - ?? 2
) = ?? 2
 ? 1 - ?? 2
= 1/4  
Problem 2: Find the equation of the ellipse whose foci are ( 2,3) , ( -2,3) and whose semi minor axis 
is of length v5. 
Solution:   Here ?? is ( 2,3) &?? '
 is ( -2,3) and ?? = v5 ? ?? '
= 4 = 2???? ? ae = 2 but  ?? 2
=
?? 2
( 1 - ?? 2
) ? 5 = ?? 2
- 4 ? ?? = 3. 
Hence the equation to major axis is ?? = 3 
Centre of ellipse is midpoint of SS' i.e. (0, 3) 
?  Equation to ellipse is 
?? 2
?? 2
+
( ?? -3)
2
?? 2
= 1 or 
?? 2
9
+
( ?? -3)
2
5
= 1 
Ans. 
Problem 3: Find the equation of the ellipse having centre at ( 1,2) , one focus at ( 6,2) and passing 
through the point ( 4,6) . 
Solution: 
With centre at ( 1,2) , the equation of the ellipse is 
( ?? -1)
2
?? 2
+
( ?? -2)
2
?? 2
= 1. It passes through the point 
( 4,6) 
? 
9
?? 2
+
16
?? 2
= 1 
Distance between the focus and the centre = ( 6 - 1) = 5 = ae 
? ?? 2
= ?? 2
- ?? 2
?? 2
= ?? 2
- 25 
Solving for ?? 2
 and ?? 2
 from the equations (i) and (ii), we get ?? 2
= 45 and ?? 2
= 20. 
Hence the equation of the ellipse is 
( ?? -1)
2
45
+
( ?? -2)
2
20
= 1 
Ans. 
Do yourself - 1: 
(i) If LR of an ellipse 
?? 2
?? 2
+
?? 2
?? 2
= 1, ( ?? < ?? ) is half of its major axis, then find its eccentricity. 
(ii) Find the equation of the ellipse whose foci are ( 4,6) &( 16,6) and whose semi-minor axis is 4 . 
(iii) Find the eccentricity, foci and the length of the latus-rectum of the ellipse ?? 2
+ 4?? 2
+ 8?? -
2?? + 1 = 0. 
2. ANOTHER FORM OF ELLIPSE: 
 
?? 2
?? 2
+
?? 2
 ?? 2
= 1, ( ?? < ?? ) 
(a) ????
'
= Minor axis = 2?? 
(b) ????
'
= Major axis = 2?? 
(c) ?? 2
= ?? 2
( 1 - ?? 2
) 
(d) Latus rectum ?? ?? '
= ?? 1
?? 1
'
=
2?? 2
?? , equation ?? = ± be 
(e) Ends of the latus rectum are: 
?? (
?? 2
 ?? , ???? ) , ?? '
(-
?? 2
 ?? , ???? ) , ?? 1
(
?? 2
 ?? , -???? ) , ?? 1
'
(-
?? 2
 ?? , -???? ) 
(f) Equation of directrix ?? = ±?? /?? 
(g) Eccentricity:?? = v1 -
?? 2
 ?? 2
 
Problem 4: The equation of the ellipse with respect to coordinate axes whose minor axis is equal to 
the distance between its foci and whose ???? = 10, will be- 
(A) 2?? 2
+ ?? 2
= 100 
(B) ?? 2
+ 2?? 2
= 100 
(C) 2?? 2
+ 3?? 2
= 80 
(D) none of these 
Solution: 
Whena > ?? 
As given 2 ?? = 2???? ? ?? = ???? 
Also 
2?? 2
?? = 10 ? ?? 2
= 5?? 
Now since ?? 2
= ?? 2
- ?? 2
?? 2
 ? ?? 2
= ?? 2
- ?? 2
 
[ From (i)] 
? 2 ?? 2
= ?? 2
 
(ii), (iii) ? ?? 2
= 100, ?? 2
= 50 
Hence equation of the ellipse will be 
?? 2
100
+
?? 2
50
= 1 ? ?? 2
+ 2?? 2
= 100 
Similarly when ?? < ?? then required ellipse is 2?? 2
+ ?? 2
= 100 
Ans. (A, B) 
3. GENERAL EQUATION OF AN ELLIPSE 
Let (a, b) be the focus ?? , and ???? + ???? + ?? = 0 is the equation of directrix. 
Let ?? ( ?? , ?? ) be any point on the ellipse. Then by definition. 
? ???? = ?????? (e is the eccentricity) ? ( ?? - ?? )
2
+ ( ?? - ?? )
2
= ?? 2
( ???? +???? +?? )
2
( ?? 2
+?? 2
)
 
Page 5


ELLIPSE 
The ellipse is a fundamental conic section with wide-ranging applications in mathematics, physics, 
and engineering. This geometric shape, defined by its unique properties, plays a crucial role in 
various fields from planetary orbits to architectural design. 
This section provides a comprehensive exploration of ellipses, starting with their standard equation 
and key definitions. We'll examine the essential components of an ellipse, including its foci, 
vertices, axes, and eccentricity. The material covers various forms of ellipse equations, methods for 
determining the position of points relative to an ellipse, and the concept of the auxiliary circle and 
eccentric angle. 
As we delve deeper, we'll investigate the parametric representation of ellipses, their interaction 
with lines, and the properties of tangents to ellipses. Through a combination of theoretical 
explanations and practical problems, this material offers a thorough grounding in the mathematics 
of ellipses, preparing the way for more advanced study and applications in related fields. 
1. STANDARD EQUATION & DEFINITION: 
Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is 
?? 2
?? 2
+
?? 2
?? 2
=
1. where ?? > ?? &?? 2
= ?? 2
( 1 - ?? 2
) ? ?? 2
- ?? 2
= ?? 2
?? 2
. where ?? = eccentricity ( 0 < ?? < 1) . 
FOCI: S = ( ???? , 0) & ?? '
= ( -???? , 0) . 
(a) Equation of directrices: 
?? =
?? ?? & ?? = -
?? ?? 
(b) Vertices: 
?? '
= ( -?? , 0) & ?? = ( ?? , 0) . 
(c) Major axis: The line segment ?? '
?? in which the foci ?? '
& ?? lie is of length 2 a & is called the major 
axis ( ?? > ?? ) of the ellipse. Point of intersection of major axis with directrix is called the 
foot of the directrix ( ?? )(±
?? ?? , 0). 
(d) Minor Axis: The y-axis intersects the ellipse in the points ?? '
= ( 0, -?? ) & ?? = ( 0, b). The line 
segment B'B of length 2 ?? ( ?? < ?? ) is called the Minor Axis of the ellipse. 
(e) Principal Axes: The major & minor axis together are called Principal Axes of the ellipse. 
(f) Centre: The point which bisects every chord of the conic drawn through it is called the centre of 
the conic. ?? = ( 0,0) the origin is the centre of the ellipse 
?? 2
?? 2
+
?? 2
?? 2
= 1. 
(g) Diameter: A chord of the conic which passes through the centre is called a diameter of the 
conic. 
(h) Focal Chord: A chord which passes through a focus is called a focal chord. 
(i) Double Ordinate: A chord perpendicular to the major axis is called a double ordinate. 
(j) Latus Rectum: The focal chord perpendicular to the major axis is called the latus rectum. 
(i) Length of latus rectum (LL') =
2 ?? 2
?? =
( ?????????? ???????? )
2
 ?????????? ???????? 
= 2?? ( 1 - ?? 2
) 
(ii) Equation of latus rectum: ?? = ± ae. 
(iii) Ends of the latus rectum are ?? (???? ,
?? 2
?? ), ?? '
(???? , -
?? 2
?? ), ?? 1
(-???? ,
?? 2
?? ) and ?? 1
(-???? , -
?? 2
?? ). 
? ???? + ?? ' ?? = 2?? = ?????????? ???????? .  
(l) Eccentricity: ?? = v1 -
?? 2
?? 2
 
Note: 
(i) The sum of the focal distances of any point on the ellipse is equal to the major Axis. Hence 
distance of focus from the extremity of a minor axis is equal to semi major axis. i.e BS = ???? . 
(ii) If the equation of the ellipse is given as 
?? 2
?? 2
+
?? 2
?? 2
= 1& nothing is mentioned, then the rule is to 
assume that ?? > ?? . 
Problem 1: If LR of an ellipse is half of its minor axis, then its eccentricity is - 
(A) 
3
2
 
(B) 
2
3
 
(C) 
v3
2
 
(D) 
v2
3
 
Solution: 
  ???? ?????????? 
2?? 2
?? = ?? ? 2 ?? = ??   ? 4?? 2
( 1 - ?? 2
) = ?? 2
 ? 1 - ?? 2
= 1/4  
Problem 2: Find the equation of the ellipse whose foci are ( 2,3) , ( -2,3) and whose semi minor axis 
is of length v5. 
Solution:   Here ?? is ( 2,3) &?? '
 is ( -2,3) and ?? = v5 ? ?? '
= 4 = 2???? ? ae = 2 but  ?? 2
=
?? 2
( 1 - ?? 2
) ? 5 = ?? 2
- 4 ? ?? = 3. 
Hence the equation to major axis is ?? = 3 
Centre of ellipse is midpoint of SS' i.e. (0, 3) 
?  Equation to ellipse is 
?? 2
?? 2
+
( ?? -3)
2
?? 2
= 1 or 
?? 2
9
+
( ?? -3)
2
5
= 1 
Ans. 
Problem 3: Find the equation of the ellipse having centre at ( 1,2) , one focus at ( 6,2) and passing 
through the point ( 4,6) . 
Solution: 
With centre at ( 1,2) , the equation of the ellipse is 
( ?? -1)
2
?? 2
+
( ?? -2)
2
?? 2
= 1. It passes through the point 
( 4,6) 
? 
9
?? 2
+
16
?? 2
= 1 
Distance between the focus and the centre = ( 6 - 1) = 5 = ae 
? ?? 2
= ?? 2
- ?? 2
?? 2
= ?? 2
- 25 
Solving for ?? 2
 and ?? 2
 from the equations (i) and (ii), we get ?? 2
= 45 and ?? 2
= 20. 
Hence the equation of the ellipse is 
( ?? -1)
2
45
+
( ?? -2)
2
20
= 1 
Ans. 
Do yourself - 1: 
(i) If LR of an ellipse 
?? 2
?? 2
+
?? 2
?? 2
= 1, ( ?? < ?? ) is half of its major axis, then find its eccentricity. 
(ii) Find the equation of the ellipse whose foci are ( 4,6) &( 16,6) and whose semi-minor axis is 4 . 
(iii) Find the eccentricity, foci and the length of the latus-rectum of the ellipse ?? 2
+ 4?? 2
+ 8?? -
2?? + 1 = 0. 
2. ANOTHER FORM OF ELLIPSE: 
 
?? 2
?? 2
+
?? 2
 ?? 2
= 1, ( ?? < ?? ) 
(a) ????
'
= Minor axis = 2?? 
(b) ????
'
= Major axis = 2?? 
(c) ?? 2
= ?? 2
( 1 - ?? 2
) 
(d) Latus rectum ?? ?? '
= ?? 1
?? 1
'
=
2?? 2
?? , equation ?? = ± be 
(e) Ends of the latus rectum are: 
?? (
?? 2
 ?? , ???? ) , ?? '
(-
?? 2
 ?? , ???? ) , ?? 1
(
?? 2
 ?? , -???? ) , ?? 1
'
(-
?? 2
 ?? , -???? ) 
(f) Equation of directrix ?? = ±?? /?? 
(g) Eccentricity:?? = v1 -
?? 2
 ?? 2
 
Problem 4: The equation of the ellipse with respect to coordinate axes whose minor axis is equal to 
the distance between its foci and whose ???? = 10, will be- 
(A) 2?? 2
+ ?? 2
= 100 
(B) ?? 2
+ 2?? 2
= 100 
(C) 2?? 2
+ 3?? 2
= 80 
(D) none of these 
Solution: 
Whena > ?? 
As given 2 ?? = 2???? ? ?? = ???? 
Also 
2?? 2
?? = 10 ? ?? 2
= 5?? 
Now since ?? 2
= ?? 2
- ?? 2
?? 2
 ? ?? 2
= ?? 2
- ?? 2
 
[ From (i)] 
? 2 ?? 2
= ?? 2
 
(ii), (iii) ? ?? 2
= 100, ?? 2
= 50 
Hence equation of the ellipse will be 
?? 2
100
+
?? 2
50
= 1 ? ?? 2
+ 2?? 2
= 100 
Similarly when ?? < ?? then required ellipse is 2?? 2
+ ?? 2
= 100 
Ans. (A, B) 
3. GENERAL EQUATION OF AN ELLIPSE 
Let (a, b) be the focus ?? , and ???? + ???? + ?? = 0 is the equation of directrix. 
Let ?? ( ?? , ?? ) be any point on the ellipse. Then by definition. 
? ???? = ?????? (e is the eccentricity) ? ( ?? - ?? )
2
+ ( ?? - ?? )
2
= ?? 2
( ???? +???? +?? )
2
( ?? 2
+?? 2
)
 
 
? ( ?? 2
+ ?? 2
) {( ?? - ?? )
2
+ ( ?? - ?? )
2
} = ?? 2
{???? + ???? + ?? }
2
 
4. POSITION OF A POINT W.R.T. AN 
ELLIPSE: 
The point ?? ( ?? 1
, ?? 1
) lies outside, inside or on the ellipse according as ; 
?? 1
 
2
?? 2
+
?? 1
 
2
 ?? 2
- 1 >< ???? = 0. 
5. AUXILLIARY CIRCLE/ECCENTRIC ANGLE: 
A circle described on major axis as diameter is called the auxiliary circle. Let ?? be a point on the 
auxiliary circle ?? 2
 +?? 2
= ?? 2
 such that QP produced is perpendicular to the ?? - axis then ?? & ?? are 
called as the CORRESPONDING POINTS on the ellipse & the auxiliary circle respectively. '  ' is 
called the ECCENTRIC ANGLE of the point ?? on 
 
 
the ellipse ( 0 = ?? < 2?? ) . 
Note that 
?? ( ???? )
?? ( ???? )
=
?? ?? =
 ???????? ?????????? ???????? 
 ???????? ?????????? ????????