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TOPIC : Conic Sections
SUBTOPIC : 1.4 Ellipse
LEARNING OBJECTIVES
(a) To define an ellipse.
(b) To determine the equation of the ellipse
with centre ( 0, 0 ) and
foci ( ± c,0 ) or ( 0, ± c).
(c) To determine all the vertices, latus rectum,
foci, major and minor axes.
An ellipse is the set of all points P(x,y)
such that the sum of its distances from
two fixed points (called foci) is a constant.
y
P(x,y)
x
F1(-c,o)
PF1+ PF2= constant
F2(c,o)
F1 and F2 are the foci
PF1  PF2  2a
 x  c
2
y 
2
 x  c
 x  c   y  2a 
2
2
x

c

y
 

 x  c   y   2a 

2

 x  c  y 

 x  c   y  4a  4a
2
2
2
1
 x  c  y  
4a
2
2
 y  2a
2
2
2
2
2
2
2
2
2
x

c

y

x

c

y
 
 
2
 x  c   y
2
2
2
2
 4a   x  c   y
2
2
2


1 2
2
2
2
2
x

c

y


x

2
xc

c

4
a

x

2
xc

c


4a
2
2

1
2
 x  c   y   4 xc  4a
4a
2
2

c
 x  c  y  a  x
a
2
2
2
c
2
 x  c   y 2  a 2  2 xc  2 x 2
a
2
c
x 2  2 xc  c 2  y 2  a 2  2 xc  2 x 2
a
2
c
x2  c2  y 2  a2  2 x2
a

2

 2
c
2
x 1  2   y  a 2  c 2
 a 
2
2

 2
a

c
2
2
2
x 

y

a

c

2
 a 
2
2
x
y
 2 2 1
2
a a c
Substitute b  a  c
2
2
2
2
2
x
y
The equation of the


1
a 2 b2
ellipse with centre at (0,0)
and foci at (-c,0) and (c,0)
a) The equation of the ellipse with centre at
(0,0) and foci at (-c,0) and (c,0)
2
2
x y
  1, where a  b  0 and
a b
2
2
b a c
2
2
2
a2 > b2
NOTE :
The major axis is along the x-axis
The vertices are at V1(-a,0) and V2(a,0)
y
V1(-a,o)
F1(-c,o)
0
Major axis
x
F2(c,o)
V2(a,o)
F1 and F2 : Two fixed points on the major
axis are called foci
V1V2 : The longer axis of length 2a is
known as major axis
y
Major axis
V4(o,b)
V1(-a,o)
F1(-c,o)
0
F2(c,o)
V2(a,o)
V3(o,-b)
Minor axis
The minor axis is along the y-axis
V3V4 : the shorter axis of length 2b is called
the minor axis
The intersection between the major axis
and minor axis is at O, the centre of ellipse
x
Latus rectum
y
V4(o,b)
P
V1(-a,o)
F1(-c,o)
Q
Major axis
0
R
F2(c,o)
V3(o,-b)
V2(a,o)
x
S
Minor axis
PQ and RS : The chord which passes through
either focus and is perpendicular to
the major axis is known as the latus rectum
2
2b
(the length of the latus rectum is
)
a
Example 1
Find the equation of the ellipse with
centre (0,0) and with the vertices at (3,0)
and (0,2)
Solution
The standard equation of an ellipse is
2
2
2
2
x
y
 1
a b
We know
a = 3 and b = 2
Equation of the ellipse
2
2
2
2
x y
 1
3 2

2
2
x y
 1
9 4
Example 2
Find the equation of an ellipse with its
centre at the origin, one of the foci is at
(3,0) and vertices is at (-4,0).
Solution
The standard equation of an ellipse is
y

V1(-4,0)

F1(-3,0)
0

F2(3,0)

V2(4,0)
x
x2
y2
 2 1
2
a
b
Hence
a=4
and c = 3
b  a  c  4  3  16 9
2
2
2
2
2
7
and the equation of the ellipse is
2
2
x
y

1
16 7
Example 3
Find the vertices, foci and state the length
of the latus rectum for the ellipse
4x2 + 16y2 = 64 .
Sketch its graph.
2
2
2
2
x y
 1
a b
Solution
4x2 + 16y2 = 64
To put the given equation in this form,
we divide each side by 64
Hence,
2
2
x
y

1
16 4
note that the major axis
is along the x-axis
Where a = 4, b = 2 and c =
16  4  12
a = 4, b = 2 and c = 12
Vertices = (4,0),(-4,0)
foci = ( 12 ,0),( - 12 ,0)
2b
The length of latus rectum =
a
y




-4  12
2
2
2  2
2
4

12
0
 -2
2

4
x
b) The equation of an ellipse with centre
at (0,0) and foci at (0,-c) and (0,c)
2
2
2
2
x y
  1, where a  b  0 and b 2  a 2  c 2
b a
a2 > b2
The major axis is along the y-axis
The vertices are at V1(0,a) and V2(0,-a)
y
V2(0,a)

F2(0,c)
x
0
F1(0,-c)

V1(0,-a)
0 : the center of ellipse
F1 and F2 : foci
V1 and V2 : major
vertices
V1V2 : the major axis
y
R

V3(-b,0)
P
V3V4 : the minor axis
V2(0,a)
 S
F2(0,c)
0

V4(b,0)
x
F1(0,-c)

Q

V1(0,-a)
PQ,RS: latus rectum
Length of latus
rectum is 2b 2
a
Example 4
Find an equation of the ellipse having
one focus at (0,2) and vertices at (0,-3)
and (0,3).Sketch the ellipse.
Solution
V2 (0,-3) and V1 (0,3)
The centre of this ellipse is at the origin.
Its major axis lies on the y-axis
a=3 and c=2
b2=a2-c2 = 9- 4 = 5

Equation of the ellipse is
2
2
x y
 1
5 9
y
3
2
5

0
 -2
 -3

5
x
Example 5
Find an equation for the ellipse that
has its centre at the origin with vertices
V (0,  5) and minor axis of length 3.
Sketch the ellipse.
Solution
a =5,
Since the minor axis is of length 3,
3
b2=a2-c2
we have b =
9
2
=25 – c2
4
91
c=
2

Equation of the ellipse is
4x y
 1
9 25
2
2
y
5
 91
2
x
0
91
2

 -5
An equation of the ellipse with centre at (0,0)
2
2
2
2
(b)
x
y
(a) x
y


1


1
2
2
b
a
a2
b2







a² > b²
c2 = a2 – b2



2b 2
Length of latus rectum =
a