Ellipses
Find the center, vertices, and foci of the ellipse with the given equation.
y2
x2
+ = 1
1)
100 36
A) Center: (0, 0); Vertices: (-10, 0), (10, 0); Foci: (-8, 0), (8, 0)
B) Center: (0, 0); Vertices: (0, -10), (0, 10); Foci: (0, -6), (0, 6)
C) Center: (0, 0); Vertices: (-10, 0), (10, 0); Foci: (-6, 0), (6, 0)
D) Center: (0, 0); Vertices: (0, -10), (0, 10); Foci: (0, -8), (0, 8)
2)
x2 y2
+ = 1
16 25
A) Center: (0, 0); Vertices: (-5, 0), (5, 0); Foci: (0, -4), (0, 4)
B) Center: (0, 0); Vertices: (0, -5), (0, 5); Foci: (-4, 0), (4, 0)
C) Center: (0, 0); Vertices: (0, -5), (0, 5); Foci: (0, -3), (0, 3)
D) Center: (0, 0); Vertices: (-5, 0), (5, 0); Foci: (-3, 0), (3, 0)
3)
(x + 3)2 (y + 4)2
+ = 1
25
9
A) Center: (-3, -4); Vertices: (-4, -8), (-4, 2); Foci: (-4, -7), (-4, 1)
B) Center: (-3, -4); Vertices: (-8, -4), (2, -4); Foci: (-6, -4), (0, -4)
C) Center: (-3, -4); Vertices: (-4, -8), (-4, 2); Foci: (-4, -6), (-4, 0)
D) Center: (-3, -4); Vertices: (-8, -4), (2, -4); Foci: (-7, -4), (1, -4)
4)
(x + 2)2 (y + 2)2
+ = 1
144
225
A) Center: (-2, -2); Vertices: (-2, -17), (-2, 13); Foci: (-14, -2), (10, -2)
B) Center: (-2, -2); Vertices: (-17, -2), (13, -2); Foci: (-2, -14), (-2, 10)
C) Center: (-2, -2); Vertices: (-17, -2), (13, -2); Foci: (-11, -2), (7, -2)
D) Center: (-2, -2); Vertices: (-2, -17), (-2, 13); Foci: (-2, -11), (-2, 7)
5) 3x2 + 8y2 = 24
A) Center: (0, 0); Vertices: B) Center: (0, 0); Vertices: C) Center: (0, 0); Vertices: D) Center: (0, 0); Vertices: -8, 0 , 8, 0 ; Foci: - 55, 0 , 55, 0
-2 2, 0 , -2 2, 0 ; Foci: - 5, 0 , 5, 0
0, -8 , 0, 8 ; Foci: 0, - 55 , 0, 55
0, -2 2 , 0, -2 2 ; Foci: 0, - 5 , 0, 5
6) 7x2 + 5y2 = 35
A) Center: (0, 0); Vertices: B) Center: (0, 0); Vertices: C) Center: (0, 0); Vertices: D) Center: (0, 0); Vertices: - 7, 0
-7, 0 , 0, - 7
0, -7 , , - 7, 0 ; Foci: - 2, 0 , 2, 0
7, 0 ; Foci: -2 6, 0 , 2 6, 0
, 0, 7 ; Foci: 0, - 2 , 0, 2
0, 7 ; Foci: 0, -2 6 , 0, 2 6
PreCalculus
Match the given graph with its equation.
7)
y
6
4
2
-8
-6
-4
-2
2
4
6
8 x
-2
-4
-6
A)
x2 y2
+ = 1
25
9
B)
x2 y2
+ = 1
10
6
C)
x2 y2
+ = 1
5
3
D)
x2 y2
+ = 1
9
25
C)
y2 x2
+ = 1
25
9
D)
y2 x2
- = 1
25
9
8)
y
6
4
2
-8
-6
-4
-2
2
4
6
8 x
-2
-4
-6
A)
x2 y2
+ = 1
25
9
B)
y2 x2
+ = 1
10
6
9)
y
6
4
2
-8
-6
-4
-2
2
4
6
8 x
-2
-4
-6
A) 9x2 + 16y 2 = 144
B) 16x2 - 9y2 = 144
C) 16x2 + 9y2 = 144
Calin M. Agut - 2012
D) 9x2 - 16y 2 = 144
PreCalculus
10)
y
4
2
-8
-6
-4
-2
2
4
6
8 x
-2
-4
-6
-8
A)
(x + 1)2 (y - 3)2
+ = 1
4
16
B)
(x + 1)2 (y - 3)2
+ = 1
16
4
C)
(x - 1)2 (y + 3)2
+ = 1
4
16
D)
(x - 1)2 (y + 3)2
+ = 1
4
16
11)
y
8
6
4
2
-8
-6
-4
-2
2
4
6
8 x
-2
-4
A)
(x - 1)2 (y + 3)2
+ = 1
4
16
B)
(x + 1)2 (y - 3)2
+ = 1
4
16
C)
(x + 1)2 (y - 3)2
+ = 1
4
16
D)
(x - 1)2 (y + 3)2
+ = 1
16
4
Calin M. Agut - 2012
PreCalculus
Graph the ellipse.
x2 y2
+ = 1
12)
25
9
16
y
12
8
4
-16 -12
-8
-4
8
12
16 x
4
8
12
16 x
4
8
12
16 x
4
-4
-8
-12
-16
13) 25(x + 1)2 + 4(y - 2)2 = 100
16
y
12
8
4
-16 -12
-8
-4
-4
-8
-12
-16
14) 4x2 + 49y2 = 196
16
y
12
8
4
-16 -12
-8
-4
-4
-8
-12
-16
Calin M. Agut - 2012
PreCalculus
Find an equation in standard form for the ellipse that satisfies the given conditions.
15) Vertices at (±10, 0) and foci at (±5 , 0)
x2
y
x
y2
x2
y2
A)
+ = 1
B)
+ = 2
C)
+ = 1
100 75
10 75
100 75
16) The vertical major axis is of length 18, and the minor axis is of length 6.
x2 y2
x2 y2
x2 y2
A)
+ = 1
B)
+ = 1
C)
+ = 1
81
9
9
3
3
9
17) The horizontal major axis is of length 18, and the minor axis is of length 8.
x2 y2
x2 y2
x2 y2
A)
+ = 1
B)
+ = 1
C)
+ = 1
9
4
81 16
4
9
18) Major axis endpoints (0, ±7), minor axis length 4
y2 x2
x2 y2
A)
+ = 1
B)
+ = 1
49
4
2
7
19) Minor axis endpoints (±2, 0), major axis length 24
x2
y2
y2
x2
A)
+ = 1
B)
+ = 1
144
4
144
4
D)
x2
y2
+ = 1
5625 10
D)
x2 y2
+ = 1
9
81
D)
x2 y2
+ = 1
16 81
C)
x2 y2
+ = 1
7
2
D)
x2 y2
+ = 1
49
4
C)
x2 y2
+ = 1
12
2
D)
x2 y2
+ = 1
2
12
D)
x2 y 2
+ = 1
25
4
20) An ellipse with intercepts (±5, 0) and (0, ±2), center at origin
x2 y 2
x2 y 2
x2 y 2
A)
+ = 1
B)
+ = 1
C)
+ = 1
5
2
2
5
4
25
Find the eccentricity of the ellipse.
21) x2 + 3y2 = 15
A)
15
10
22) 47x2 + y2 = 47
2162
A)
47
B)
2 3
15
C)
15
12
D)
i 2
3
B)
47
4 138
C)
47
46
D)
4 138
47
Calin M. Agut - 2012
Answer Key
Testname: 11_ELLIPSES
1) A
2) C
3) D
4) D
5) B
6) C
7) A
8) C
9) C
10) C
11) C
12)
14)
16
12
8
4
-16 -12
-4
4
-8
-12
y
-16
12
8
4
-8
-4
4
8
12
16 x
4
8
12
16 x
-4
-8
15) C
16) D
17) B
18) A
19) B
20) D
21) D
22) A
-12
-16
13)
16
y
12
8
4
-16 -12
-8
-4
16
-16 -12
y
-8
-4
-4
-8
-12
-16
Calin M. Agut - 2012
8
12
16 x