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1.4 Ellipse1.4 Ellipse

• Another conicsection formedby a plane intersecting acone

• Ellipse formed when

90

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Definition:

An ellipse is defined as the set of points in a plane such that the sum of the distances from P to two fixed points is a constant. The two fixed points are the foci.

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Graph of an EllipseGraph of an Ellipse

Note various parts of an ellipse

Note various parts of an ellipse

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The equation of an ellipse with centre (0,0)

and foci )0,( c

x

y

cF2 (-c.0) F1 (c,0)V2(-a,o) V1(a,0)

M1(0,b)

M2(0,-b)

G

H

J

K

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We summarized the properties of the ellipse with the horizontal major axis as,

12

2

2

2

b

y

a

xa > b >0

Vertices : )0,( aMajor axis : horizontal, length 2a

Minor axis : vertical, length 2b

Foci : where c2=a2-b2

Latus rectum : vertical length

)0,( c

a

b22

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The equation of an ellipse with center (0,0) and foci ),0( c

x

y

c

F1(0,c)

F2(0,-c)

V2(0,-b)

V1(0,b)

M1(0,a)M2(0,-a)

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We summarised the properties of this second form of ellipse as follow:-

12

2

2

2

b

y

a

xb > a >0

Vertices :

Major axis : vertical, length 2b

Minor axis : horizontal, length 2a

Foci where c2=b2-a2

Latus rectum: vertical length

),0( b

),0( c

b

a22

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The equation of an ellipse with centre (h,k) and foci

1)()(

2

2

2

2

b

ky

a

hx a > b >0

Vertices : ),( kah

Major axis : horizontal, length 2a

Minor axis : vertical, length 2b

Foci : where c2=a2-b2

Latus rectum : vertical length

),( kch

a

b22

),( kch

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1)()(

2

2

2

2

b

ky

a

hxb > a >0

Vertices :

Major axis : vertical, length 2b

Minor axis : horizontal, length 2a

Foci where c2=b2-a2

Latus rectum: vertical length

),( bkh

),( ckh

b

a22

The equation of an ellipse with center (h,k) and foci ),( ckh

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Example 1

Find the equation for the ellipse that has its centre at the origin with vertices V (0,± 7) and Foci ( 0,± 2 ).

• Solution

The standard equation of an ellipse is 12

2

2

2

b

y

a

x

where 222 abc ; 22 ab

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• Since the vertices are ( 0,± 7 ), we conclude that a = 7. Since the Foci are (0,±2), we have c = 2 .

= 22 + 72

= 4 + 49

= 53

222 cab

and equation of the ellipse is 15349

22

yx

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Example 2Example 2

Find the 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

The standard equation of an ellipse is 12

2

2

2

b

y

a

x

where 222 abc ; 22 ab

Since the vertices are ( 0,± 5 ), we conclude that b = 5.

Since the minor axis is of length 3, we have 2

3a

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And equation of the ellipse is And equation of the ellipse is

1259

4 22

yx

0,2

3

0,

2

3

(0, 5)

(0, –5)

0

y

x

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Example 3Example 3

Find the focus and equation of the ellipse

with centre (0,0) vertices at (2,0) and

(0,4).

Solution

From the above

2a and 4b

22 24 c

12c

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• Equation of ellipse is

and Foci is ( 0, ) and

1416

22

xy

12 12,0

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Example 4Example 4

• Find the centre an vertices of the minor axis and the

Foci of the ellipse .

Solution

The equation of an ellipse is 1)()(

2

2

2

2

b

ky

a

hx

For equation 14

)1(

9

)3( 22

yx

,

14

)1(

9

)3( 22

yx

,

The centre of the ellipse is 1,3 ; b = 2 ,a = 3 .

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• Vertices of the minor axis are and

• Foci of the ellipse are and

Since , c2 = a2 - b2

= 9 – 4

= 5

3,3 1,3

1 ,53 1 ,53

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Example 5Example 5

• Write the equation of the ellipse that has vertices at and and Foci at and 5,3 5,7 5,1 5,5

Solution The vertices and foci are on the same horizontal line

5y . The equation of the ellipse is ,

1)()(

2

2

2

2

b

ky

a

hx

Where a > bThe centre of the ellipse is at the midpoint of the major axes

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• h = and k = 22

73

5

2

)5(5

The distance between the centre 5,2 and vertex 5,7

is 5 units ; thus 5a.

The distance between the centre ( 2,-5) and focus

( 5,-5) is 3 units, thus c = 3

222

bac

,

222 cab

= 925

= 16

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116

)5(

25

)2( 22

yx

400525216 22 yx

4002510254416 22 yyxx

40062525025646416 22 yyxx

The equation of the ellipse is

0289250256416 22 yyxx

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Example 6Example 6

• Find the equation of an ellipse with centre ( 3,1 ) and the major axis running parallel with the y axis. The length of the major axis is 10 units and the minor axis is 6 units.

• Sketch the ellipse.

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SolutionSolution

The equation for an ellipse with centre ( h,k ) and the major axis running parallel with the y axis is

where ( b² > a² )

• The length of the major axis is 10 units and the minor axis is 6 units.

• We get 2b = 10 , 2a = 6 b = 5 , a = 3

1)1()3( 2

2

2

b

y

a

x

435 22 C

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• The equation of the ellipse is 125

)1(

9

)3( 22

yx

A

.

.

y

x

F1 (3,5)

B

F2 (-3,-3)

DC (3,1)

E

.

(3,6)(3,6)

(-3,-4)

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Example 7Example 7

• Find the equation of ellipse with vertices ( 8,5 ) and ( 10,1 ) with centre ( 8,k ).

Solution

• Sketching the vertices of the ellipse given.

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(10,1)

(8,5)

(x,y)

(x1,y1)

x

y

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• We get the centre of ellipse is ( 8,1) , k = 1 x = 8, x1 = 6, y1 = 1

• So equation of ellipse is 14

)1(

2

)8(2

2

2

yx

4

)8( 2x1

16

)1( 2

y

3y

+

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Example 8Example 8• Sketch the graph of the equation,

Solution:

• Complete the squares for the expressions

16( x2 + 4x +4 ) + 9( y2 – 2y + 1 ) = 71 + (16)(4) + (9)(1)

16 ( x + 2 )2 + 9 ( y – 1 )2 = 144

0711864916 22 yxyx

711864916 22 yxyx

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• The equation is an ellipse with centre

c ( -2,1) and a = 3, b = 4

c2 = b2 – a2

= 16 – 9

= 7

• c = ±

• Foci are

116

)1(

9

)2( 22

yx

7

71,2

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(-2,5)

(1,1)(-5,1)

(-2,1)

71,21 F

71,22 Fx

y

Graph for equation 116

)1(

9

)2( 22

yx