PreCalc_Notes_10.2_Ellipses123456890.ppt

•Ellipse – a set of points P in a plane such that
the sum of the distances from P to 2 fixed
points (F1 and F2) is a given constant K.
•Fixed points - foci

•Major axis – the segment that contains the
foci and has its endpoints on the ellipse.
•Endpoints of major axis are vertices
•Midpoint of major axis is the center of the
ellipse.
•Minor axis – perpendicular to major axis at
the center
•Endpoints of minor axis are co-vertices

2 2 2 2
2 2 2 2
x
1 1
x major y
2a length 2a
y minor x
2b length
x y y
a b b a
   
2b
( a, 0) vertices (0, a)
(0, b) co-vertices ( b, 0)
( ,0) foci (0, c)
c
 
 
 

a > b
Length from center to foci = c
c2
= a2
– b2
Foci are always on major axis
Write on
your note
card!

Write an equation if a vertex is (0, -4) and a
co-vertex is (3, 0) and the center is (0, 0)
4
a 
3
b 
2
2
1
9
1
6
x y
 

•Write an equation if centered at the origin
and is 20 units wide and 10 units high.
10
a 
5
b 
2
2
2
100
1
5
y
x
 

Remember…
•Equation of a circle:
•Center (h,k)
2 2 2
( ) ( )
x h y k r
   
2 2 2 2
2 2 2 2
( ) ( ) ( ) ( )
1 1
(0 a, 0) vertices (0, a)
(0, 0 b) co-vertices (0 b, 0)
(0 ,0) foci
x y x y
a b b a
c
   
 
 
 (0, 0 c)


2 2 2 2
2 2 2 2
( ) ( ) ( ) ( )
1 1
(h a, k) vertices (h, k a)
(h, k b) co-vertices (h b, k)
(h , )
x h y k x h y k
a b b a
c k
   
   
 
 
 foci (h, k c)

Write this down on your note
card.
Remember, a > b

Eccentricity of an Ellipse
•Measures how ‘circular’ the ellipse is (describes
the shape of the ellipse.)
•If e is close to 0 then foci are near center and
more round.
•If e is close to 1 then foci are far from center
and ellipse is elongated.
, so must be between 0 and 1 (0<e<1)
c
e e
a

Write on your note card!

Find center, foci, length of major and
minor, vertices and co-vertices and
graph.
2 2
( 1) ( 2)
1
20 4
x y
 
 
:(1, 2)
Center 
2 2 2
:
F c b
o a
ci   4
c 
( , )
h c k
  
1 4, 2
 
: 2
major a 4 5

: 2
minor b 4

:( , )
v h
ertice a
s k

 
1 2 5, 2
 
:( , )
co vertice h
s k b


 
1, 2 2
 
   
1,0 & 1, 4

 
 
5, 2
& 3, 2

 

Find center, foci, length of major and minor,
vertices and co-vertices and graph.
x2
+ 4y2
– 6x – 16y – 11 = 0
2
2
1
6 1
4( 4 )
y
x x y

 

2
2
4( 4 4) 1
6 9 9
11 6
x y
x y   
  
 
2 2
4(
( 3 36
2)
)
x y  
 
2 2
(
( 3)
36
2
9
1
)
y
x 
 


Find center, foci, length of major and minor,
vertices and co-vertices.
25x2
+ 4y2
- 150x + 40y + 225 = 0
2 2
2
4( 10 )
) 5
6 2
25( y
x y
x 
 

2
2
4( 10 25
25( 6 9) 225
2 1
2 00
5
)
x x y y
 
 
 

 
2 2
4(
2 1
)
5 0
( 3) 0
5
y
x  
 
2 2
( 3)
4
( )
25
1
5
y
x 




Write an equation of the ellipse
with the given characteristics:
•Center at (-4, 1), vertical major axis 18 units
long, minor axis 12 units long
9
a 
6
b 
2 2
(
( 1)
8
6 1
)
1
4
3
y
x 
 


Write an equation of the ellipse with the given
characteristics:
•Foci at (-1, 0) and (1, 0) and a = 4
Foci c

2 2 2
c a b
 
2 2 2
1 4 b
 
15
b 
:(0,0)
Center
2
2
1
16
1
5
y
x
 

characteristics:
•Foci at (3, 5) and (1, 5) and eccentricity 1/4
:(2,5)
Center
c
e
a

1
4

Foci c

2 2 2
c a b
 
2 2 2
1 4 b
 
15
b 
2 2
(
( 5)
1
6 5
)
1
2
1
y
x 
 


characteristics:
•Tangent to the x and y-axes and has center at
(4, -7)
7
a 
4
b 
2 2
(
( 7)
4
6 9
)
1
4
1
y
x 
 


What happens if the denominators are
equal????
IT’S A CIRCLE!!!!!!
IT’S A CIRCLE!!!!!!

PreCalc_Notes_10.2_Ellipses123456890.ppt

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PreCalc_Notes_10.2_Ellipses123456890.ppt