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The Ellipse

3.4.2


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An ellipse is 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.

r

Axis

The Ellipse

Major AxisMin

o

Focus 1 Focus 2

PointPF1 + PF2 = constant

3.4.2


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The Ellipse

Standard Form Equation of an Ellipse

The general form for the standard form equation of an ellipse is

3.4.2


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The standard form of an ellipse centred at the origin with the major

axis oflength 2a along thex-axis and a minor axis oflength 2b alongthey-axis, is:

x2

2 ++++

y2

2 ==== 1

The Standard Forms of the Equation of the Ellipse

3.4.3


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The standard form of an ellipse centred at the origin with

the major axis oflength 2a along they-axis and a minor axisoflength 2b along thex-axis, is:

2 2

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

2 ++++ 2 ==== 1

3..4


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The Ellipse

3.4.2


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The Ellipse

3.4.2


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The Pythagorean Property

b aa2 =b2 +c2

2 = 2 - 2F

1(-c, 0)

2(c, )

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)

3.4.5


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The standard form of an ellipse centred at any point (h,k)with the major axis oflength 2a parallel to thex-axis and

a minor axis oflength2b parallel to they-axis, is:

(x h)

2

2 ++++ (y k)

2

2 ==== 1


(h,k)

3.4.6


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(x h)2

b2

+ (y k)2

a2

=1


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

with the major axis oflength 2a parallel to they-axis and

a minor axis oflength 2b parallel to thex-axis, is:

(h, k)

3.4.7


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The general form of the ellipse is:

Ax2 + Cy2 +Dx +Ey +F = 0

A x C> 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)2

32 ++++ (y ++++ 2)

2

52 ==== 1

x2 8x ++++ 16

9++++

y2 ++++ 4y ++++ 4

25==== 1

25(x2 8x ++++ 16) ++++ 9(y

2++++ 4y ++++ 4) ==== 225

25x2 200x ++++ 400 ++++ 9y

2++++ 36y ++++ 36 ==== 225

25x2

+ 9y2

- 200x + 36y + 211 = 0

[ ]225


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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 length of the major axis is 8.

The len th of the minor axis is 6.

Finding the Centre, Axes, and Foci

b a

x y2 2

16 91++++ ====a)

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

= 16 - 9

= 7

3.4.9

c

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.

Finding the Centre, Axes, and Foci

b

c

a

x y2 2

9 41++++ ====

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

c2 =a2 - b2= 32 - 22

= 9 - 4

= 5

3.4.10c ==== 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

2 ++++y2

2 ==== 1The length of the major axis is 16 soa = 8.

The length of the minor axis is 8 sob = 4.

x

82 ++++

y

42 ==== 1

x2

64++++

y2

16==== 1 Standard form

x2

64++++

y2

16

==== 1[[[[ ]]]]

6464

x2 + 4y2 = 64

x2 + 4y2 - 64 = 0General form

3.4.11


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b)The length of the major axis is 12 soa = 6.

The length of the minor axis is 6 sob = 3.

Finding the Equation of the Ellipse With Centre at (0, 0)

xb

2 ++++ ya

2 ==== 1

x2

32++++

y2

62==== 1

x2

9++++

y2

36==== 1 Standard form

x2

9++++

y2

36

==== 1[[[[ ]]]]

3636

4x2 +y2 = 364x2 +y2 - 36 = 0 General

form

3.4.12

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