Unit #4 Conics
The Ellipse
An ellipse is the set of all points in a plane whose distances
from two fixed points in the plane, the foci, is constant.
Minor Axis
The line through the foci is called
the focal or major axis
Focus 1
Point
Major Axis
Focus 2
PF1 + PF2 = constant
When the circle is wider than it is taller this is called a
horizontal shape.
When the circle is taller than it is wider is called a vertical
shape.
The Standard Forms of the Equation of the Ellipse [cont’d]
The standard form of an ellipse centred at any point (h, k)
with the major axis of length 2a parallel to the x-axis and
a minor axis of length 2b parallel to the y-axis, is:
(x  h)
(y  k)

1
2
2
a
b
2
2
(h, k)
This is called a
what kind of
shape?
The Standard Forms of the Equation of the Ellipse [cont’d]
The standard form of an ellipse centred at any point (h, k)
with the major axis of length 2a parallel to the y-axis and
a minor axis of length 2b parallel to the x-axis, is:
(x  h) (y  k)

1
2
2
b
a
2
2
(h, k)
This is called a
what kind of
shape?
The Pythagorean Property
b
F1(-c, 0)
a
c F (c, 0)
2
a2 = b2 + c2
b2 = a2 - c2
c2 = a2 - b2
Length of major axis: 2a
Length of minor axis: 2b
Vertices: (a, 0) and (-a, 0)
Foci: (-c, 0) and (c, 0)
Finding the General Form of the Ellipse
The general form of the ellipse is:
Ax2 + Cy2 + Dx + Ey + F = 0
A x C > 0 and A ≠ C
The general form may be found by expanding the
standard form and then simplifying:
[
(x  4)2 (y  2)2

1
2
2
3
5
x2  8x  16 y 2  4y  4

1
9
25
]
25(x 2  8x  16)  9(y 2  4 y  4)  225
25x 2  200x  400  9 y 2  36y  36  225
25x2 + 9y2 - 200x + 36y + 211 = 0
225
Finding the Center, Vertices, and Foci
State the coordinates of the vertices, the coordinates of the foci,
and the lengths of the major and minor axes of the ellipse,
defined by each equation.
x
y
 1
a)
The center of the ellipse is (0, 0).
16 9
2
2
b
a
c
Since the larger number occurs under the x2,
the major axis lies on the x-axis.
a2=16; a=4; Since the major axis is 2a, it equals 8.
b2=9; b =3; so the length of the minor axis is 2b = 6.
The coordinates of the vertices are (4, 0) and (-4, 0).
To find the coordinates of the foci, use the Pythagorean property:
c2 = a2 - b2
= 42 - 32
The coordinates of the foci are:
= 16 - 9
(  7,0 ) and ( 7,0 )
=7
c 7
You Do: Finding the Center, Vertices, and Foci
b) 4x2 + 9y2 = 36
x y
 1
9 4
2
2
a
b
c
The centre of the ellipse is (0, 0).
Since the larger number occurs under the x2,
the major axis lies on the x-axis.
The length of the major axis is 6.
The length of the minor axis is 4.
The coordinates of the vertices are (3, 0) and (-3, 0).
To find the coordinates of the foci, use the Pythagorean property.
c2 = a2 - b2
= 32 - 22
=9-4
=5
c 5
The coordinates of the foci are:
(  5 ,0 ) and ( 5 ,0 )
Finding the Equation of the Ellipse With Centre at (0, 0)
a) Find the equation of the ellipse with centre at (0, 0),
foci at (5, 0) and (-5, 0), a major axis of length 16 units,
and a minor axis of length 8 units.
Since the foci are on the x-axis, the major axis is the x-axis.
x2 y 2
2  2  1
a
b
x2 y 2
2  2  1
8
4
x2 y 2

1
64 16
The length of the major axis is 16 so a = 8.
The length of the minor axis is 8 so b = 4.
64
64
x2 y 2 
64  16  1


Standard form
x2 + 4y2 = 64
x2 + 4y2 - 64 = 0 General form
Finding the Equation of the Ellipse
b)
Centre at (0, 0)
The length of the major axis is 12 on the y-axis
so a = 6.
The length of the minor axis on the x-axis is 6
so b = 3.
x2 y 2
2  2  1
b
a
x2 y 2
2  2  1
3
6
x2 y 2

1
9 36
36
x2 y 2 
36
9  36  1


4x2 + y2 = 36
4x2 + y2 - 36 = 0
Standard form
General
form
Finding the Equation of the Ellipse With Centre at (h, k)
b)
The major axis is parallel to the x-axis and
(-3, 2)
has a length of 12 units, so a = 6.
The minor axis is parallel to the y-axis and
has a length of 6 units, so b = 3.
The centre is at (-3, 2), so h = -3 and k = 2.
(x  h) 2 (y  k)2

1
2
2
a
b
(x  (3)) 2 (y  2)2

1
2
2
6
3
(x  3)2 (y  2)2

1
36
9
Standard form
(x + 3)2 + 4(y - 2)2 = 36
(x2 + 6x + 9) + 4(y2 - 4y + 4) = 36
x2 + 6x + 9 + 4y2 - 16y + 16 = 36
x2 + 4y2 + 6x - 16y + 25 = 36
General form
x2 + 4y2 + 6x - 16y - 11 = 0
Convert from General form to Standard Form
Example) x2 + 4y2 - 2x + 8y - 11 = 0
x2 + 4y2 - 2x + 8y - 11 = 0
(x2 - 2x ) + (4y2 + 8y) - 11 = 0
1
4
1
1 + _____
(x2 - 2x + _____)
+ 4(y2 + 2y + _____)
= 11 + _____
(x - 1)2 + 4(y + 1)2 = 16
Since the larger number
( x  1) ( y  1)
h= 1
occurs under the x2, the

1
k = -1
major axis is parallel to
16
4
2
the x-axis.
c2 = a2 - b2
= 42 - 22
= 16 - 4
= 12
c  12
c2 3
2
a= 4
b= 2
The centre is at (1, -1).
The major axis, parallel to the x-axis,
has a length of 8 units.
The minor axis, parallel to the y-axis,
has a length of 4 units.
The foci are at
(1  2 3, 1) and (1  2 3, 1).
Sketching the Graph of the Ellipse [cont’d]
(x  1)2 ( y  1)2

1
16
4
x2 + 4y2 - 2x + 8y - 11 = 0
Centre (1, -1)
(1- 2 3, - 1)
(1  2 3,  1)
(1, -1)
F2
F1
c 2 3
c 2 3
Analysis of the Ellipse
b) 9x2 + 4y2 - 18x + 40y - 35 = 0
9x2 + 4y2 - 18x + 40y - 35 = 0
(9x2 - 18x ) + (4y2 + 40y) - 35 = 0
1
25 = 35 + _____
9 + _____
100
9(x2 - 2x + _____)
+ 4(y2 + 10y + _____)
9(x - 1)2 + 4(y + 5)2 = 144
Since the larger number
occurs under the y2, the
major axis is parallel to
the y-axis.
c2 = a2 - b2
= 62 - 42
= 36 - 16
= 20
c  20
c2 5
( x  1) ( y  5 )

1
16
36
2
2
h=
k=
a=
b=
1
-5
6
4
The centre is at (1, -5).
The major axis, parallel to the y-axis,
has a length of 12 units.
The minor axis, parallel to the x-axis,
has a length of 8 units.
The foci are at:
(1, 5  2 5 ) and (1, 5  2 5 )
Sketching the Graph of the Ellipse [cont’d]
9x2
+
4y2
- 18x + 40y - 35 = 0
(x  1)2 ( y  5)2

1
16
36
F1 (1,  5  2 5 )
c 2 5
c 2 5
F2 (1, -5 - 2 5)
General Effects of the Parameters A and C
Ax2 + Cy2 + Dx + Ey + F = 0
When A ≠ C, and A x C > 0, the resulting
conic is an ellipse.
If | A | > | C |, it is a vertical ellipse.
If | A | < | C |, it is a horizontal ellipse.
The closer in value A is to C, the closer
the ellipse is to a circle. This distance is called the
Eccentricity.
Definition of Eccentricity of an Ellipse
The eccentricity of an ellipse is
c
a b
e 
a
a
2
2
Pages 652-653
A 2, 4, 6, 7-10, 16, 22, 24, 26, 28, 30, 34,
38, 40, 45-48, 50