Transcript Warm-up

Warm-up
Write the equation of the parabola given the focus is at (-2,0) and the vertex
is at the origin.
Write the equation of the circle with radius 3√7
Ellipses
Definition
An ellipse is the set of all points in a plane such that
of an
the sum of the distances from the foci is constant.
Ellipse
y
(b, 0)

a


(a, 0) F(c, 0)
b a
x
O c


F(c, 0) (a, 0)

minor axis
(b, 0) major axis
The length of the major
axis is 2a.
The length of the minor
axis is 2b.
Notice that a > b.
Ellipses
An elongated circle that has a major axis(longer) with endpoints called
vertices and a minor axis (shorter) with endpoints called co-vertices.
There are 2 equations depending on whether the ellipse is longer
vertically or horizontally.
x
2
a
2

y
2
b
2
1
Horizontal Ellipse
x
2
b
2

y
2
a
2
1
Vertical Ellipse
What is the difference?
Yes the a switched places: 2 facts a > b and the equation
must = 1.
Ellipses
a2 is always larger
and it is always
under the
coordinate that has
the vertices and the
focus.
There are three things you will find :
1) The vertices (a)
2) The co-vertices (b)
3) The foci
(c)
You find c by the equation c2 = a2 - b2
x
2
a
2

y
2
b
2
1
x
2
b
2

y
2
a
2
1
vertices (  a , 0)
vertices (0,  a )
co  vertices (0,  b )
co  vertices (  b , 0)
foci (  c , 0)
foci (0,  c )
Ellipses
Given an equation find the vertices, co-vertices, and foci and draw the
ellipse
2
2
x

25
y
1
16
10
a  25
b  16
a5
b4
2
2
c  25  16
2
5
c
-10
-5
c 9
2
c3
a
b
-5
-10
5
10
vertices (  5, 0)
co  vertices (0,  4)
foci (  3, 0)
Ellipses
Given an equation find the vertices, co-vertices, and foci and draw the
ellipse
2
2
x

9
y
1
64
10
c
-5
-10
b3
c  55
2
c
5
a
a8
2
b
-5
b 9
2
c  64  9
5
-10
a  64
2
10
55
vertices (0,  8)
co  vertices (  3, 0)
foci (0,  55 )
Ellipses
4x
2
 49 y
 196
2
4x
2
2
 196
196
x
2

49
10
 49 y
y
2
 1
4
c  49  4
2
a  49 b  4 c 2  4 5
2
5
a7
a
-10
-5
5
a
-5
-10
10
2
b2
vertices (  7, 0)
c
co  vertices (0,  2)
foci (  45 , 0)
45
Writing an equation
What if you were given a vertex and a co-vertex and center at the origin could you
find the equation?
Sure you could square both of them and write the equation making sure you put them
under the correct coordinate
Vertex ( 3,0)
Co-vertex (0,1)
vertex (0,-6)
co-vertex ( 4,0)
a  3, a  9
a  6, a  36
2
b  1, b  1
b  4, b  16
2
x
2
x
2
9

y
1
1
2
2
2
16

y
2
36
1
Writing an equation
What if you were given a vertex or co-vertex and a focus and center at the origin
could you find the equation?
Sure you could square both of them then use c2 = a2 - b2 to find the other then
write the equation making sure you put them under the correct coordinate
Vertex ( 3,0)
Focus (-2,0)
a  3, a  9
2
c  2, c  4
2
b 945
2
x
2
9

y
2
5
1
focus (0,3√2)
co-vertex ( 4,0)
c  3 2 , c  9  2  18
2
b  4, b  16
2
a  18  16  34
2
x
2
16

y
2
34
1