1. Ellipses
Date: ____________

2. Ellipses Standard Equation of an Ellipse Center at (0,0) x2 a2 y2 b2 +
= 1
(a, 0) x O
(0, –b)

3. Horizontal Major Axis Vertical Major Axis Co-Vertices Vertices
a2 < b2
a2 > b2

4. For example,
An ellipse is the set of all points P in a plane such that the sum of the distances from P to two fixed points, F1 and F2, called the foci, is a constant. P P P F1 F2 2a
F1P + F2P = 2a

5. Horizontal Major Axis:y x2 a2 y2 b2 +
= 1
(0, b)
(a, 0)
(–a, 0)
a2 > b2
a2 – b2 = c2 x O
F1(–c, 0)
F2 (c, 0)
(0, –b)
length of major axis: 2a length of minor axis: 2b
Distance from midpoint and foci: c

6. length of major axis: 2b length of minor axis: 2a
Vertical Major Axis: y x2 a2 y2 b2 +
= 1
(0, b)
F1 (0, c)
b2 > a2
(–a, 0)
(a, 0) x O
b2 – a2 = c2
F2(0, –c)
length of major axis: 2b length of minor axis: 2a
(0, –b)
Distance from midpoint and foci: c

7. Write an equation of an ellipse in standard form with the center at the origin and with the given vertex and co-vertex.
(4,0), (0,3)
Vertices : (4,0)
Co-Vertices: (0,3)
(-4,0)
(0,-3)
So a = 4
So b = 3
a² = 16
b² = 9 x2 16 y2 9 +
= 1

8. Find an equation of an ellipse for the given height and width with the center at (0,0)
h = 32 ft, w = 16 ft
32 ft
Distance b is from the center is 16
16 ft
Distance a is from the center is 8 x2 64 y2
256 +
= 1
a = 8
a² = 64
b = 16
b² = 256

9. Find the foci and graph the ellipse.x2 25 y2 9 +
= 1 x y
a2 = 25
b2 = 9
a = ±5
b = ±3
(0, 3)
25 – 9 = c2
(–5, 0)
(–4, 0)
(5, 0)
16 = c2
(4, 0)
±4 = c
(0,-3)

10. Graph the ellipse. Find the foci.x2 9 y2 25 +
= 1 x y
a2 = 9
b2 = 25
(0, 5)
a = ±3
b = ±5
(0,4)
(–3, 0)
(3, 0)
b2 – a2 = c2
25 – 9 = c2
(0,-4)
16 = c2
±4 = c
(0,-5)

11. Write an equation of an ellipse for the given foci and co-vertices.
Foci: (±5,0), co-vertices: (0,±8)
Horizontal axis
Since c = 5 and b = 8
c² = 25 and b² = 64 x2 a2 y2 b2 +
= 1
a2 – b2 = c2
a2 – 64 = 25
+ 64
+ 64 x2 89 y2 64 +
= 1
a2 = 89

12. Translated Ellipses Standard Equation of an Ellipse Center at (h,k)
(x – h)2 a2
(y – k)2 b2 +
= 1 y
(h, k+b)
(h+a, k)
(h,k)
(h, k–b)
(h–a, k) x

13. Write an equation of the translation.
Center = (2,-5)
h = 2
k = -5
Horizontal major axis of length 12, minor axis of length 8.
Length of major axis is 2a
Length of minor axis is 2b
2a = 12
2b = 8
a = 6
b = 4
a2 = 36
b2 = 16
(x – 2)2 36
(y + 5)2 16 +
= 1
(x – h)2 a2
(y – k)2 b2 +
= 1

14. Find the foci for the ellipse.
4x2 + 9y2 – 16x +18y – 11 = 0
4x2 – 16x + 9y2 + 18y = 11
4(x2 – 4x + ____) + 9(y2 + 2y + ___) =11 4 1
+16 +9
4(x – 2)2 + 9(y + 1)2 = 36 36
(x – 2)2 9
(y + 1)2 4 +
= 1

15. (x – 2)2 9 (y + 1)2 4 = 1 Foci = (2 + 2.2,-1) Foci = (2 – 2.2,-1)
Center = (2,-1)
Foci = (4.2,-1)
and = (-0.2, -1)
a2 = 9
b2 = 4
a2 > b2
Horizontal Axis
a2 – b2 = c2
9 – 4 = c2
5 = c2
±2.2 ≈ c