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Ellipses - Calculus - Exercise, Exercises of Calculus

National Institute of Industrial EngineeringCalculus

This file contains some problems related calculus. Some hints to the given problems are: Ellipses, Vertices, Given Equation, Center, Given Graph, Equation, Graph the Ellipse, Standard Form, Satisfies, Given Conditions

Typology: Exercises

2011/2012

Uploaded on 12/31/2012

aparijita
aparijita 🇮🇳

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bg1
Ellipses
Findthecenter,vertices,andfocioftheellipsewiththegivenequation.
1) x2
100 +y2
36 =1
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
16 +y2
25 =1
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
25 +(y+4)2
9=1
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
144 +(y+2)2
225 =1
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:-8,0,8,0 ;Foci:-55,0 ,55,0
B) Center:(0,0);Vertices:-22,0 ,-22,0 ;Foci:-5,0 ,5,0
C) Center:(0,0);Vertices:0,-8 ,0,8 ;Foci:0,-55 ,0,55
D) Center:(0,0);Vertices:0,-22,0,-22;Foci:0,-5 ,0,5
6) 7x2+5y2=35
A) Center:(0,0);Vertices:-7,0 ,-7,0 ;Foci:-2,0 ,2,0
B) Center:(0,0);Vertices:-7,0 ,7,0 ;Foci:-26,0 ,26,0
C) Center:(0,0);Vertices:0,-7 ,0,7 ;Foci:0,-2 ,0,2
D) Center:(0,0);Vertices:0,-7 ,0,7 ;Foci:0,-26,0,26
pf3
pf4
pf5

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Ellipses

Find the center, vertices, and foci of the ellipse with the given equation.

  1. 100 x2 + y2 36 = 1

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)

  1. x 16
  • y 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)

  1. (x^ + 25 3)2 + (y^ + 9 4)2= 1

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)

  1. (x^ +^ 2) 144
  • (y^ +^ 2) 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)

  1. 3x2^ + 8y2^ = 24 A) Center: (0, 0); Vertices: - 8, 0 , 8, 0 ; Foci: - 55, 0 , 55, 0 B) Center: (0, 0); Vertices: - 2 2, 0 , - 2 2, 0 ; Foci: - 5, 0 , 5, 0 C) Center: (0, 0); Vertices: 0, - 8 , 0, 8 ; Foci: 0, - 55 , 0, 55 D) Center: (0, 0); Vertices: 0, - 2 2 , 0, - 2 2 ; Foci: 0, - 5 , 0, 5

  2. 7x2^ + 5y2^ = 35 A) Center: (0, 0); Vertices: - 7, 0 , - 7, 0 ; Foci: - 2, 0 , 2, 0 B) Center: (0, 0); Vertices: - 7, 0 , 7, 0 ; Foci: - 2 6, 0 , 2 6, 0 C) Center: (0, 0); Vertices: 0, - 7 , 0, 7 ; Foci: 0, - 2 , 0, 2 D) Center: (0, 0); Vertices: 0, - 7 , 0, 7 ; Foci: 0, - 2 6 , 0, 2 6

Match the given graph with its equation.

-8 -6 -4 -2 2 4 6 8 x

y 6 4 2

-8 -6 -4 -2 2 4 6 8 x

y 6 4 2

A) x 25

  • y 9

= 1 B) x 10

  • y 6

= 1 C) x 5

  • y 3

= 1 D) x 9

  • y 25

-8 -6 -4 -2 2 4 6 8 x

y 6 4 2

-8 -6 -4 -2 2 4 6 8 x

y 6 4 2

A) x 25

  • y 9

= 1 B) y 10

  • x 6

= 1 C) y 25

  • x 9

= 1 D) y 25

  • x 9

-8 -6 -4 -2 2 4 6 8 x

y 6 4 2

-8 -6 -4 -2 2 4 6 8 x

y 6 4 2

A) 9x2^ + 16y2^ = 144 B) 16x2^ - 9y2^ = 144 C) 16x2^ + 9y2^ = 144 D) 9x2^ - 16y2^ = 144

Graph the ellipse.

  1. x 9
  • y 25

-16 -12 -8 -4 4 8 12 16 x

16 y 12 8 4

-16 -12 -8 -4 4 8 12 16 x

16 y 12 8 4

  1. 25(x + 1)2^ + 4(y - 2)2^ = 100

-16 -12 -8 -4 4 8 12 16 x

16 y 12 8 4

-16 -12 -8 -4 4 8 12 16 x

16 y 12 8 4

  1. 4x2^ + 49y2^ = 196

-16 -12 -8 -4 4 8 12 16 x

16 y 12 8 4

-16 -12 -8 -4 4 8 12 16 x

16 y 12 8 4

Find an equation in standard form for the ellipse that satisfies the given conditions.

  1. Vertices at (±10, 0) and foci at (±5 , 0)

A) x 100

  • y 75

= 1 B) x 10

  • y 75

= 2 C) x 100

  • y 75

= 1 D) x 5625

  • y 10
  1. The vertical major axis is of length 18, and the minor axis is of length 6.

A) x 81

  • y 9

= 1 B) x 9

  • y 3

= 1 C) x 3

  • y 9

= 1 D) x 9

  • y 81
  1. The horizontal major axis is of length 18, and the minor axis is of length 8.

A) x 9

  • y 4

= 1 B) x 81

  • y 16

= 1 C) x 4

  • y 9

= 1 D) x 16

  • y 81
  1. Major axis endpoints (0, ±7), minor axis length 4

A) y2 49 + x2 4 = 1 B) x2 2 + y2 7 = 1 C) x2 7 + y2 2 = 1 D) x2 49 + y2 4 = 1

  1. Minor axis endpoints (±2, 0), major axis length 24

A) x 144

  • y 4

= 1 B) y 144

  • x 4

= 1 C) x 12

  • y 2

= 1 D) x 2

  • y 12
  1. An ellipse with intercepts (±5, 0) and (0, ±2), center at origin

A) x 5

  • y 2

= 1 B) x 2

  • y 5

= 1 C) x 4

  • y 25

= 1 D) x 25

  • y 4

Find the eccentricity of the ellipse.

  1. x2^ + 3y2^ = 15

A) 1015 B)^2 153 C) 1215 D) i^32

  1. 47x2^ + y2^ = 47

A) 2162 47

B) 47

C) 47

D)^4