ELLIPSE – a conic section formed by the intersection of a right circular cone and a plane.

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ELLIPSE

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ELLIPSE

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Center is at ( h , k )

ELLIPSE

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Center is at ( h , k )

Center is at ( 0 , 0 )

Standard Form :

022 cbyaxyx

ELLIPSE - graphs

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When a2 > b2

-x +x

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-y

ELLIPSE - graphs

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When a2 > b2

Major axis-x +x

+y

-y

ELLIPSE - graphs

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When a2 > b2

Major axis-x +x

+y

-y

Minor axis

ELLIPSE - graphs

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When a2 > b2

Major axis-x +x

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Minor axis

Major axis vertices

ELLIPSE - graphs

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ky

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When a2 > b2

Major axis-x +x

+y

-y

Minor axis

Major axis vertices Minor axis vertices

ELLIPSE - graphs

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When a2 > b2

-x +x

+y

-yFoci - fixed coordinate points inside the ellipse

- used to create the ellipse

- the distance from one of the foci, to ANY point

on the ellipse, to the other foci is equal

- to find the foci

Foci

2222 OR abbac

ELLIPSE - graphs

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ky

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hx

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When a2 > b2

-x +x

+y

-yFoci - fixed coordinate points inside the ellipse

- used to create the ellipse

- the distance from one of the foci, to ANY point

on the ellipse, to the other foci is equal

- the green distance = the black distance

ELLIPSE - graphs

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b

ky

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hx

b

y

a

x

When b2 > a2

-x +x

+y

-y

ELLIPSE - graphs

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b

ky

a

hx

b

y

a

x

When b2 > a2

-x +x

+y

-y

Major axis

ELLIPSE - graphs

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b

ky

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hx

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x

When b2 > a2

-x +x

+y

-y

Major axis

Minor axis

ELLIPSE - graphs

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ky

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hx

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When b2 > a2

-x +x

+y

-y

Minor axis vertices

Major axis vertices

ELLIPSE - graphs

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When b2 > a2

-x +x

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-y

Foci

When working with ellipses, we will always find the following :

Center ( h , k )

a = √a2 Major Axis vertices ( x , y ), ( x , y )

b = √b2 Minor Axis vertices ( x , y ) , ( x , y )

c = Foci vertices ( x , y )

“h” is ALWAYS adjusted by “a”

“k” is ALWAYS adjusted by “b”

The Foci ALWAYS lies on the major axis

NOTE : I’d write these parameters down somewhere, the test problems are EXACTLY like these examples that you are about to see…hint, hint

22 ba

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( h , k )

Major Axis vertices ( x , y ), ( x , y )

Minor Axis vertices ( x , y ) , ( x , y )

Foci vertices ( x , y )

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( x , y ), ( x , y )

Minor Axis vertices ( x , y ) , ( x , y )

Foci vertices ( x , y )

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( x , y ), ( x , y )

Minor Axis vertices ( x , y ) , ( x , y )

Foci vertices ( x , y )

b > a , y axis is major

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( x , y ), ( x , y )

Minor Axis vertices ( x , y ) , ( x , y )

Foci vertices ( x , y )

b > a , y axis is major

(x)

(y)

(y)

The purple letters show what will be adjusted in the major and minor axis from the center

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( x , y ), ( x , y )

Minor Axis vertices ( x , y ) , ( x , y )

Foci vertices ( x , y )

b > a , y axis is major

(x)

(y)

(y)

The purple letters show what will be adjusted in the major and minor axis from the center

Major axis – x stays the same, y is adjusted by ± b

Minor axis – y stays the same, x is adjusted by ± a

±3 ±4

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( 0 , y ), ( 0 , y )

Minor Axis vertices ( x , y ) , ( x , y )

Foci vertices ( x , y )

(x)

(y)

(y)

Major axis – x stays the same, y is adjusted by ± b

Minor axis – y stays the same, x is adjusted by ± a

±3 ±4

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( 0 , y ), ( 0 , y )

Minor Axis vertices ( x , 0 ) , ( x , 0 )

Foci vertices ( x , y )

(x)

(y)

(y)

Major axis – x stays the same, y is adjusted by ± b

Minor axis – y stays the same, x is adjusted by ± a

±3 ±4

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( 0 , 4 ), ( 0 , -4 )

Minor Axis vertices ( x , 0 ) , ( x , 0 )

Foci vertices ( x , y )

(x)

(y)

(y)

Major axis – x stays the same, y is adjusted by ± b

Minor axis – y stays the same, x is adjusted by ± a

±3 ±4

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( 0 , 4 ), ( 0 , -4 )

Minor Axis vertices ( 3 , 0 ) , ( -3 , 0 )

Foci vertices ( x , y )

(x)

(y)

(y)

Major axis – x stays the same, y is adjusted by ± b

Minor axis – y stays the same, x is adjusted by ± a

±3 ±4

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( 0 , 4 ), ( 0 , -4 )

Minor Axis vertices ( 3 , 0 ) , ( -3 , 0 )

Foci vertices ( x , y )

(x)

(y)

(y)

- the Foci is adjusted by ± c

- in this case, x stays the same, y is adjusted by ± c ( ±√7)

±3 ±4

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( 0 , 4 ), ( 0 , -4 )

Minor Axis vertices ( 3 , 0 ) , ( -3 , 0 )

Foci vertices ( 0 , y )

(x)

(y)

(y)

- the Foci is adjusted by ± c

- in this case, x stays the same, y is adjusted by ± c ( ±√7)

±3 ±4

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( 0 , 4 ), ( 0 , -4 )

Minor Axis vertices ( 3 , 0 ) , ( -3 , 0 )

Foci vertices ( 0 , 0 ± √7 )

(x)

(y)

(y)

- the Foci is adjusted by ± c

- in this case, x stays the same, y is adjusted by ± c ( ±√7)

±3 ±4

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( 0 , 4 ), ( 0 , -4 )

Minor Axis vertices ( 3 , 0 ) , ( -3 , 0 )

Foci vertices ( 0 , 0 ± √7 )

(x)

(y)

(y)

±3 ±4

To graph the Ellipse, plot your center, and your major & minor vertices, then sketch a smooth curve through your points.

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( 0 , 4 ), ( 0 , -4 )

Minor Axis vertices ( 3 , 0 ) , ( -3 , 0 )

Foci vertices ( 0 , 0 ± √7 )

(x)

(y)

(y)

±3 ±4

To graph the Ellipse, plot your center, and your major & minor vertices, then sketch a smooth curve through your points.

EXAMPLE : Find all vertice points, foci points, and graph the ellipse

1169

22

yx a = 3

b = 4

c = 7916

Center ( 0 , 0 )

Major Axis vertices ( 0 , 4 ), ( 0 , -4 )

Minor Axis vertices ( 3 , 0 ) , ( -3 , 0 )

Foci vertices ( 0 , 0 ± √7 )

(x)

(y)

(y)

±3 ±4

To graph the Ellipse, plot your center, and your major & minor vertices, then sketch a smooth curve through your points.

EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

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yx

EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

1

25

3

9

9 22

yx a = 3

b = 5

c = 4

1st find a, b, and c

EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

1

25

3

9

9 22

yx a = 3

b = 5

c = 4

Next find the center…Center ( h , k )

Major Axis vertices ( x , y ), ( x , y )

Minor Axis vertices ( x , y ) , ( x , y )

Foci vertices ( x , y )

EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

1

25

3

9

9 22

yx a = 3

b = 5

c = 4

Next find the center…Center ( - 9 , 3 )

Major Axis vertices ( x , y ), ( x , y )

Minor Axis vertices ( x , y ) , ( x , y )

Foci vertices ( x , y )

EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

1

25

3

9

9 22

yx a = 3

b = 5

c = 4

Next find the major / minor vertices…

b2 > a2 so y is major, x is minor

Center ( - 9 , 3 )

Major Axis vertices ( x , y ), ( x , y )

Minor Axis vertices ( x , y ) , ( x , y ) Foci vertices ( x , y )

EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

1

25

3

9

9 22

yx a = 3

b = 5

c = 4

Major, change y by ± b

Center ( - 9 , 3 )

(y) Major Axis vertices ( x , y ), ( x , y )

(x) Minor Axis vertices ( x , y ) , ( x , y ) (y) Foci vertices ( x , y )

Minor, change x by ± a

± 3 , ±5

EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

1

25

3

9

9 22

yx a = 3

b = 5

c = 4

Center ( - 9 , 3 )

(y) Major Axis vertices ( - 9 , 8 ), ( - 9 , - 2 )

(x) Minor Axis vertices ( - 6 , 3 ) , ( - 12 , 3 ) (y) Foci vertices ( x , y )

± 3 , ±5

Major, change y by ± b

EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

1

25

3

9

9 22

yx a = 3

b = 5

c = 4

Center ( - 9 , 3 )

(y) Major Axis vertices ( - 9 , 8 ), ( - 9 , - 2 )

(x) Minor Axis vertices ( - 6 , 3 ) , ( - 12 , 3 ) (y) Foci vertices ( x , y )

± 3 , ±5

Minor, change x by ± a

EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

1

25

3

9

9 22

yx a = 3

b = 5

c = 4

Center ( - 9 , 3 )

(y) Major Axis vertices ( - 9 , 8 ), ( - 9 , - 2 )

(x) Minor Axis vertices ( - 6 , 3 ) , ( - 12 , 3 ) (y) Foci vertices ( x , y )

± 3 , ±5

Foci is on major, change y by ± c

EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

1

25

3

9

9 22

yx a = 3

b = 5

c = 4

Center ( - 9 , 3 ) ±4

(y) Major Axis vertices ( - 9 , 8 ), ( - 9 , - 2 )

(x) Minor Axis vertices ( - 6 , 3 ) , ( - 12 , 3 ) (y) Foci vertices ( - 9 , 3 ± 4 )

± 3 , ±5

Foci is on major, change y by ± c

EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

1

25

3

9

9 22

yx a = 3

b = 5

c = 4

Center ( - 9 , 3 )

Major Axis vertices ( - 9 , 8 ), ( - 9 , - 2 )

Minor Axis vertices ( - 6 , 3 ) , ( - 12 , 3 ) Foci vertices ( - 9 , 3 ± 4 )

GRAPH – 1st plot center, then plot major & minor vertices, then sketch your ellipse.

EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

1

25

3

9

9 22

yx a = 3

b = 5

c = 4

Center ( - 9 , 3 )

Major Axis vertices ( - 9 , 8 ), ( - 9 , - 2 )

Minor Axis vertices ( - 6 , 3 ) , ( - 12 , 3 ) Foci vertices ( - 9 , 3 ± 4 )

GRAPH – 1st plot center, then plot major & minor vertices, then sketch your ellipse.

EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

1

25

3

9

9 22

yx a = 3

b = 5

c = 4

Center ( - 9 , 3 )

Major Axis vertices ( - 9 , 8 ), ( - 9 , - 2 )

Minor Axis vertices ( - 6 , 3 ) , ( - 12 , 3 ) Foci vertices ( - 9 , 3 ± 4 )

GRAPH – 1st plot center, then plot major & minor vertices, then sketch your ellipse.