Section 8.3
Ellipses
Parabola
Hyperbola
Circle
Ellipse
Ellipse:
Besides having the two foci, an ellipse also has a major and minor
axis, vertices at the end of the major axis and center point where
the two axes cross.
Standard Equations for an Ellipse
Major axis Parallel to x - axis
Center
=
(0,
0)
2
2
x
y
Vertices (a, 0), (- a, 0)
+
=
1
Minor Intercepts (0, b), (0, -b)
a2
b2
2 = a2 - b2 Foci (c, 0), (- c, 0)
c
a>b>0
Major Axis = 2a Minor Axis = 2b
(0, b)
(a,0)
V
b
F
F
a
V
(- a, 0)
(0, 0)
(- c , 0)
(0, - b)
(c, 0)
Standard Equations for an Ellipse
Major axis parallel to y - axis
x2
y2
+
=
1
a>b>0
b2
a2
Center = (0, 0)
Vertices (0, a), (0, - a)
b
Minor Intercepts
(-b,0)
(b, 0), (- b, 0)
Major Axis = 2a
Minor Axis = 2b
F
c2 = a2 - b2
Foci (0, c), (0, - c)
(0,- a,)
V (0,a)
(0,c)
F
(0, 0) (b,0)
a
(0,-c)
V
Ellipse
Sketch, Find Foci, Length of Minor and Major Axis For
Center at the origin.
3
x2 y2
+
=1
16 9
a2 = 16 b2 = 9 | |- 4| | | -| 7
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a= 4 b= 3
c2 = a2 - b2 = 16 - 9 = 7
-3
c= 7
Vertices = (4, 0) & (- 4, 0)
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7
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4
Minor intercepts = (0, 3) & (0,- 3)
Foci = (7, 0) & (- 7, 0)
Maj. Axis=2·a=2(4)=8 Min. Axis=2·b=2(3)=6
Ellipse
Sketch, Find Foci, Length of Minor and Major Axis For
9
Center at the origin.
65
x2 y2
+
=1
16 81
a2
b2
-4
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= 81
= 16
4
a= 9 b= 4
- 65
c2 = a2 - b2 = 81 - 16 = 65
-9
c =  65
Vertices = (0, 9) & (0, - 9)
Minor intercepts = (4,0) & (- 4,0)
Foci = (0,  65) & (0, -  65)
Maj. Axis=2·a=2(9)=18 Min. Axis=2·b=2(4)=8
Graph the Ellipse
16x 2  y 2  16
x2 y2

1
1 16
b2  1
b  1
Needs to be set
equal to 1.
a 2  16
a  4
Vertices: (0,-4) and (0,4)
Minor Intercepts: (-1,0) and (1,0)
Find the equation of the
Foci: (-1,0) and (1,0)
ellipse
Vertices: (-3,0) and (3,0)
Therefore a = 3 and c = 1
c2  a2  b2
1  9  b2
 8  b2
8  b2
x2 y2

1
9
8
Ellipse
Find an equation of an ellipse in the form
x2 y2
+ 2 =1
2
a
b
1. When Major axis is on x-axis
Major axis length = 32 Minor axis length = 30
Therefore,
b = 30 ÷ 2 = 15
a = 32 ÷ 2 = 16
b2 = 225
a2 = 256
x2
y2
+
=1
256 225
Ellipse
Find an equation of an ellipse in the form
x 2 y2
+ 2 =1
2
b
a
2.
Major axis on y-axis
Major axis length = 16
Distance from Foci to Center = 7
Therefore, c = 7
a = 16 ÷ 2 = 8 c2 = a2 – b2
a2 = 64
b2 = a2 – c2 = 64 – 49 = 15
x2
y2
+
=1
15
64
Find the equation of the
ellipse in the form below
x2 y2
+ 2 =1
2
a
b
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if thee center is
the origin.
a = 10
b=6
a2 = 100
b2 = 36
x2 y2
+
=1
100 36
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Translations
Ellipses translate just like circles and
parabolas do…by using h and k in the
standard equation.
( x  h) ( y  k )

1
2
2
a
b
2
2
This is for a horizontal major axis, switch a and b for a
vertical major axis…if your equation isn’t in this form
you will need to complete the square to make it so…
Graph the ellipse
x  12   y  32
9
1
1
Center: (-1,3)
Major axis parallel to x-axis
a2  9
b2  1
a3
b 1
Place a point
3 units right
and left of
center
2
2
2
Place a point c  a  b
c 2  9 1
1 unit above
and below the c 2  8
c 8
center.
c  2 .8
Foci are about 2.8
units to the left and
right of center.
Graph the ellipse
4x  32x  y  2 y  57  0
2
4x
4x
4x
2
2
2
2
 32x y  2 y   57
 8x y  2 y   57
 8x  16 y  2 y  1  57  416  11
2
2
2
4x  4   y 1  8
2
x  4
2
2
2

y  1

2
8
1
x  4
2
2
b2  2
b  1 .4
Place 2 points
1.4 unit right
and left of
center

y  1

2
8
1
a2  8
a  2 .8
Place 2
points 2.8
units up and
down from
center
Major axis is parallel to
the y-axis
Center is (-4,1)
Write the equation of the ellipse
Foci: (2,-2) and (4,-2)
Vertices: (0,-2) and (6,-2)
Center is halfway between the
vertices so the point (3,-2)
We know a = 3 and c = 1
1 9b
 8  b2
8  b2
2
Plug into standard form:
x  h2   y  k 2
1
x  32   y  22
1
a
2
9
b
2
8
Write the equation of the ellipse
Major axis vertical with length of 6 and minor
axis length of 4 centered at (1,-4)
2a  6
a3
2b  4
b2
x  h 
2
b
2
1

y  4

1
a
x  1
2
4

y  k

2
2
2
9