Download - ELLIPSE – a conic section formed by the intersection of a right circular cone and a plane
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
+y
-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
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Minor axis
ELLIPSE - graphs
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ky
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When a2 > b2
Major axis-x +x
+y
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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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ky
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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
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a
x
When b2 > a2
-x +x
+y
-y
ELLIPSE - graphs
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b
ky
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hx
b
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a
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When b2 > a2
-x +x
+y
-y
Major axis
ELLIPSE - graphs
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ky
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When b2 > a2
-x +x
+y
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Major axis
Minor axis
ELLIPSE - graphs
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When b2 > a2
-x +x
+y
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Minor axis vertices
Major axis vertices
ELLIPSE - graphs
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When b2 > a2
-x +x
+y
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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
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yx
EXAMPLE : Find all vertice points, foci points, and graph the ellipse
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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
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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
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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
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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
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25
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.