34the Ellipse

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MATHPOWERTM12, WESTERN EDITION 3.4.1

3.4

Chapter 3 Conics


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An ellipseis the locus of all points in a plane such that

the sum of the distances from two given points in the plane,

the foci, is constant.

Major AxisMinorA

xis

Focus 1 Focus 2

PointPF1 + PF2= constant

3.4.2

The Ellipse


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The standard formof an ellipse centred at the origin with the major

axis of length 2aalong thex-axisand a minor axis of length 2balong

they-axis, is:

x2

a2 y2

b2 1

3.4.3

The Standard Forms of the Equation of the Ellipse


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The standard formof an ellipse centred at the origin with

the major axis of length 2aalong they-axisand a minor axis

of length 2balong thex-axis, is:

x2

b2 y2

a2 1

3.4.4

The Standard Forms of the Equation of the Ellipse [contd]


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F1(-c, 0) F2(c, 0)

The Pythagorean Property

b

c

aa2= b2+ c2

b2 = a2 -c2

c2= a2 -b2

Length of major axis: 2aLength of minor axis: 2b

Vertices: (a, 0) and (-a, 0)

Foci: (-c, 0) and (c, 0)

3.4.5


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The standard formof an ellipse centred at any point (h, k)

with the major axis of length 2aparallel to thex-axisand

a minor axis of length 2bparallel to they-axis, is:

(x h) 2a

2 (yk)2b

2 1(h, k)

3.4.6



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(x h)2

b2

(y k)2

a2 1

(h, k)


3.4.7

The standard formof an ellipse centred at any point (h, k)

with the major axis of length 2aparallel to they-axisand

a minor axis of length 2bparallel to thex-axis, is:


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The general formof the ellipse is:

Ax2+ Cy2 +Dx +Ey+ F= 0

AxC> 0 andA C

The general form may be found by expanding the

standard form and then simplifying:

Finding the General Form of the Ellipse

3.4.8

(x 4)23

2 (y 2)25

2 1x2 8x16

9

y2 4y425

125(x

2 8x16)9(y2 4 y4) 22525x

2 200x400 9 y2 36y 3622525x2+ 9y2- 200x+ 36y+ 211 = 0

[ ]

225

Fi di h C A d F i


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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.

The centre of the ellipse is (0, 0).

Since the larger number occurs under thex2,

the major axis lies on thex-axis.

The coordinates of the vertices are (4, 0) and (-4, 0).

The length of the major axis is 8.

The length of the minor axis is 6.

To find the coordinates of the foci, use the Pythagorean property:

c2= a2- b2

= 42- 32

= 16 - 9

= 7

Finding the Centre, Axes, and Foci

3.4.9

bc

a

x y2 2

16 91a)

c 7

The coordinates of the foci are:

( , )7 0 and ( , )7 0


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b) 4x2+ 9y2= 36

The centre of the ellipse is (0, 0).

Since the larger number occurs under thex2,

the major axis lies on thex-axis.

The coordinates of the vertices are (3, 0) and (-3, 0).

The length of the major axis is 6.

The length of the minor axis is 4.

To find the coordinates of the foci, use the Pythagorean property.

c2= a2- b2

= 32- 22

= 9 - 4

= 5

3.4.10

Finding the Centre, Axes, and Foci

b

c

a

x y2 2

9 41

c 5

The coordinates of the foci are:

( , )5 0 and ( , )5 0


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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 thex-axis, the major axis is thex-axis.

x2

a2 y2

b2 1 The length of the major axis is 16 so a= 8.The length of the minor axis is 8 so b= 4.

x2

82 y2

42 1

x2

64 y

2

16 1 Standard formx2

64 y2

16

1

6464

x2+ 4y2= 64

x2+ 4y2- 64 = 0 General form

3.4.11


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a) Find the equation for the ellipse with the centre at (3, 2),

passing through the points (8, 2), (-2, 2), (3, -5), and (3, 9).

The major axis is parallel to they-axis and has a length of 14 units, so a= 7.

The minor axis is parallel to thex-axis and has a length of 10 units, so b= 5.

The centre is at (3, 2), so h= 3 and k= 2.

(x h) 2b

2 (yk)2a

2 1(x 3)2

52 (y2)2

72 1

(x 3)225

(y2)249

1 49(x- 3)2+ 25(y- 2)2= 1225

49(x2- 6x+ 9) + 25(y2- 4y+ 4) = 1225

49x2- 294x+ 441 + 25y2- 100y+ 100 = 1225

49x2+ 25y2-294x- 100y+ 541 = 1225

49x2+ 25y2-294x- 100y- 684 = 0

Standard form

General form3.4.13

Finding the Equation of the Ellipse With Centre at (h, k)

(3, 2)


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(-3, 2)

b)The major axis is parallel to thex-axis and

has a length of 12 units, so a = 6.

The minor axis is parallel to they-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) 2a

2 (yk)2b

2 1(x (3))2

62 (y2)2

32 1

(x 3)236

(y2)29

1 (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

x2+ 4y2+ 6x- 16y- 11 = 0

Standard form

General form3.4.14

Finding the Equation of the Ellipse With Centre at (h, k)


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Find the coordinates of the centre, the length of the major and

minor axes, and the coordinates of the foci of each ellipse:

F1(-c, 0) F2(c, 0)

b

c

a

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)

Recall:

a

P

PF1+ PF2= 2a

c

3.4.15

Analysis of the Ellipse


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a) x2+ 4y2- 2x+ 8y- 11 = 0

x2+ 4y2- 2x+ 8y- 11 = 0

(x2- 2x) + (4y2+ 8y) - 11 = 0

(x2- 2x+ _____) + 4(y2+ 2y+ _____) = 11 + _____ + _____1 11 4

(x- 1)2+ 4(y+ 1)2= 16

h =

k =

a =b =

1

-1

42

Since the larger number

occurs under thex2, the

major axis is parallel to

thex-axis.

c2= a2- b2

= 42- 22

= 16 - 4

= 12

The centre is at (1, -1).

The major axis, parallel to thex-axis,

has a length of 8 units.

The minor axis, parallel to they-axis,


The foci are at

3.4.16

Analysis of the Ellipse [contd]

( ) ( )x y 116

1

41

2 2

c 12c 2 3

( , )1 2 3 1 and ( , ).1 2 3 1

Sk hi h G h f h Elli [ d]


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x2+ 4y2- 2x+ 8y- 11 = 0(x1)2

16 ( y1)2

4 1

F1F

2

c2 3(1 2 3, 1)(1- 2 3, - 1)

c2 3

3.4.17

Sketching the Graph of the Ellipse [contd]

Centre (1, -1)

(1, -1)


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b) 9x2+ 4y2- 18x+ 40y- 35 = 0

9x2+ 4y2- 18x+ 40y- 35 = 0

(9x

2

- 18x) + (4y

2

+ 40y) - 35 = 09(x2- 2x+ _____) + 4(y2+ 10y+ _____) = 35 + _____ + _____1 25 9 100

9(x- 1)2+ 4(y+ 5)2= 144

h =

k =a =

b =

1

-56

4

Since the larger number

occurs under they2, the

major axis is parallel tothey-axis.

c2= a2- b2

= 62- 42

= 36 - 16= 20

The centre is at (1, -5).

The major axis, parallel to they-axis,has a length of 12 units.

The minor axis, parallel to thex-axis,


The foci are at:

3.4.18

Analysis of the Ellipse

( ) ( )x y 116

5

361

2 2

c 20c 2 5

( , )1 5 2 5 and ( , )1 5 2 5

Sk hi h G h f h Elli [ d]


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9x2+ 4y2- 18x+ 40y- 35 = 0 (x1)216

( y5)236

1F1

F2

c2 5

c2 5

(1, 52 5 )

(1, -5 - 2 5)

3.4.19

Sketching the Graph of the Ellipse [contd]


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Graphing an Ellipse Using a Graphing Calculator

(x1)216

( y1)24

1

y 16 (x1)2

4 1

y 16 (x 1)2

4

1

y 16 (x 1)2

4 1

y12

16 (x1)2

4

y1 16 (x1)24

(x- 1)2+ 4(y+ 1)2= 16

4(y+ 1)2= 16 - (x- 1)2

3.4.20


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3 4 22

Pages 150-152

A 1-20

B 21, 23, 25, 33,36, 39, 40

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34the Ellipse