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
Co-Vertices
Vertices

4. Graph the ellipse. Find the vertices and co-vertices.x2 25 y2 9 +
= 1 x y
a2 = 25
b2 = 9
a = ±5
b = ±3
(0, 3)
(–5, 0)
(5, 0)
Horizontal Major Axis
(0,-3)
Vertices: (–5, 0) and (5,0)
Co-vertices:(0, 3) and (0,-3)

5. Graph the ellipse. Find the vertices and co-vertices.x2 9 y2 25 +
= 1 x y
a2 = 9
b2 = 25
(0, 5)
a = ±3
b = ±5
(–3, 0)
(3, 0)
Vertical Major Axis
Vertices: (0,5) and (0,-5)
(0,-5)
Co-vertices: (-3,0) and (3,0)

6. Translated Ellipses Standard Equation of an Ellipse Center at (h,k)
(x – h)2 a2
(y – k)2 b2 +
= 1 y
(0, k+b)
(h+a, 0)
(h,k)
(0, k–b)
(h–a, 0) x

7. Graph the ellipse Center = (2,-5) (x – 2)2 36 (y + 5)2 16 + = 1
b2 = 16
a = ±6
b = ±4
Horizontal Major Axis
Vertices: (8,-5) and (-4,-5)
Co-vertices: (2,-1) and (2,-9)

8. Graph the ellipse Center = (-3,-1) (x + 3)2 25 (y + 1)2 81 + = 1
b2 = 81
a = ±5
b = ±9
Vertical Major Axis
Vertices: (-3,8) and (-3,-10)
Co-vertices: (-8,-1) and (2,-1)

9. Write the equation of the ellipse in standard form. Graph
Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices.
4x2 + 25y2 = 100 x y
100 x2 25 y2 4 +
= 1
Center = (0,0)
a2 = 25
b2 = 4
a = ±5
b = ±2
Vertices: (-5,0) and (5,0)
Co-vertices: (0,2) and (0,-2)

10. Write the equation of the ellipse in standard form. Graph
Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices.
25x2 + 3y2 = 75 75 x y x2 3 y2 25 +
= 1
Center = (0,0)
a2 = 3
b2 = 25
a ≈ ±1.73
b = ±5
Vertices: (0,5) and (0,-5)
Co-vertices: (-1.73,0) and
(1.73,0)

11. x2 – 4x + ____ + 9(y2 + 6y + ___) = -49 +4 +81
Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices.
x2 + 9y2 – 4x + 54y + 49 = 0
x2 – 4x + 9y2 + 54y = -49 4 9
x2 – 4x + ____ + 9(y2 + 6y + ___) = -49 +4
+81
(x – 2)2 + 9(y + 3)2 = 36 36
(x – 2)2 36
(y + 3)2 4 +
= 1

12. Center = (2,-3) (x – 2)2 36 (y + 3)2 4 + = 1 a2 = 36 b2 = 4 a = ±6
Vertices: (-4,-3) and (8,-3)
Co-vertices: (2,-1) and (2,-5)

13. 4(x2 + 6x + ____) + y2 – 4y + ___ = -36 9 4 +36 + 4
Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices.
4x2 + y2 +24x – 4y + 36 = 0
4x2 + 24x + y2 – 4y = -36
4(x2 + 6x + ____) + y2 – 4y + ___ = -36 9 4
+36
+ 4
4(x + 3)2 + (y – 2)2 = 4 4
(x + 3)2 1
(y – 2)2 4 +
= 1

14. (x + 3)2 1 (y – 2)2 4 = 1 Center = (-3,2) + a2 = 1 b2 = 4 a = ±1
Vertices: (-3,4) and (-3,0)
Co-vertices: (-4,2) and (-2,2)

15. 4(x2 – 4x + ____) + 9(y2 + 2y + ___) =11 4 1 +16 +9
Write the equation of the ellipse in standard form. Graph. Find the center, vertices, and co-vertices.
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

16. (x – 2)2 9 (y + 1)2 4 = 1 Center = (2,-1) + a2 = 9 b2 = 4 a = ±3
Vertices: (5,-1) and (-1,-1)
Co-vertices: (2,1) and (2,-3)

17. 9.4 Ellipses
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

18. Horizontal Major Axis:
9.4 Ellipses 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

19. length of major axis: 2b length of minor axis: 2a
9.4 Ellipses
Vertical Major Axis: y x2 a2 y2 b2 +
= 1
(0, b)
F1 (0, c)
(–a, 0)
(a, 0)
b2 > a2
b2 – a2 = c2 x O
F2(0, –c)
length of major axis: 2b length of minor axis: 2a
(0, –b)

20. Find the foci. x2 25 y2 9 + = 1 25 – 9 = c2 16 = c2 ±4 = c y (–4, 0) x
(4, 0)

21. Find the foci. x2 9 y2 25 + = 1 25 – 9 = c2 16 = c2 ±4 = c y (0,4) x
(0,-4)

22. Find the foci. x2 100 y2 36 + = 1 100 – 36 = c2 64 = c2 ±8 = c y
(–8, 0)
(8, 0)

23. Find the foci. (x – 4)2 16 (y – 3)2 25 + = 1 25 – 16 = c2 9 = c2
(4, 6)
9 = c2
±3 = c
(4, 0)

24. Find the foci. (x + 1)2 4 (y + 2)2 16 + = 1 16 – 4 = c2 12 = c2
(-1,1.5)
±3.5 ≈ c
(-1,-5.5)