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Parabolas - Calculus - Exercise, Exercises of Calculus

Aligarh Muslim UniversityCalculus

These are the notes of Exercise of Calculus. Key important points are: Parabolas, Vertex, Focus, Directrix, Focal Width, Standard Form, Equation of the Parabola, Testname

Typology: Exercises

2012/2013

Uploaded on 02/11/2013

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Parabolas
Findthevertex,focus,directrix,andfocalwidthoftheparabola.
1) x2=28y
A) Vertex:(0,0);Focus:(0,7);Directrix:y= -7;Focalwidth:28
B) Vertex:(0,0);Focus:(0,-7);Directrix:x= -7;Focalwidth:112
C) Vertex:(0,0);Focus:(7
,
0);Directrix:x=7;Focalwidth:7
D) Vertex:(0,0);Focus:(7
,
0);Directrix:y=7;Focalwidth:112
2) -1
40 x2=y
A) Vertex:(0,0);Focus:(0,-10);Directrix:y=10;Focalwidth:40
B) Vertex:(0,0);Focus:(-20
,
0);Directrix:x=10;Focalwidth:160
C) Vertex:(0,0);Focus:(0,10);Directrix:y= -10;Focalwidth:10
D) Vertex:(0,0);Focus:(0,-10);Directrix:y=10;Focalwidth:160
3) x=8y2
A) Vertex:(0,0);Focus:0,1
32 ;Directrix:y=-1
32 ;Focalwidth:32
B) Vertex:(0,0);Focus:1
32 ,0 ;Directrix:x=1
32 ;Focalwidth:32
C) Vertex:(0,0);Focus:1
8,0 ;Directrix:x=-1
8;Focalwidth:0.13
D) Vertex:(0,0);Focus:1
32 ,0 ;Directrix:x=-1
32 ;Focalwidth:0.13
4) y2=8x
A) Vertex:(0,0);Focus:(0,2);Directrix:y= -2;Focalwidth:2
B) Vertex:(0,0);Focus:(2
,
0);Directrix:x= -2;Focalwidth:8
C) Vertex:(0,0);Focus:(2
,
0);Directrix:x= -2;Focalwidth:32
D) Vertex:(0,0);Focus:(2
,
2);Directrix:x=2;Focalwidth:32
5) (y-8)2=16(x-2)
A) Vertex:(2
,
8);Focus:(6
,
8);Directrix:x= -2;Focalwidth:16
B) Vertex:(8
,
2);Focus:(8
,
18;Directrix:y= -14;Focalwidth:16
C) Vertex:(2
,
8);Focus:(18
,
8);Directrix:x= -14;Focalwidth:16
D) Vertex:(8
,
2);Focus:(8
,
6);Directrix:y= -2;Focalwidth:4
6) (x-8)2=16(y-6)
A) Vertex:(6
,
8);Focus:(10
,
8);Directrix:x=4;Focalwidth:4
B) Vertex:(6
,
8);Focus:(22
,
8);Directrix:x= -8;Focalwidth:16
C) Vertex:(8
,
6);Focus:(8
,
10);Directrix:y=2;Focalwidth:16
D) Vertex:(-8
,
-6);Focus:(-8
,
10);Directrix:y= -22;Focalwidth:16
Findthestandardformoftheequationoftheparabola.
7) Vertexattheorigin,focusat(0,9)
A) y=1
9x2B) y2=36x C) y2=9x D) y=1
36 x2
pf3
pf4
pf5

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Parabolas

Find the vertex, focus, directrix, and focal width of the parabola.

  1. x2^ = 28y A) Vertex: (0, 0); Focus: (0, 7); Directrix: y = -7; Focal width: 28 B) Vertex: (0, 0); Focus: (0, - 7); Directrix: x = -7; Focal width: 112 C) Vertex: (0, 0); Focus: (7, 0); Directrix: x = 7; Focal width: 7 D) Vertex: (0, 0); Focus: (7, 0); Directrix: y = 7; Focal width: 112

x2^ = y

A) Vertex: (0, 0); Focus: (0, - 10); Directrix: y = 10; Focal width: 40 B) Vertex: (0, 0); Focus: (-20, 0); Directrix: x = 10; Focal width: 160 C) Vertex: (0, 0); Focus: (0, 10); Directrix: y = -10; Focal width: 10 D) Vertex: (0, 0); Focus: (0, - 10); Directrix: y = 10; Focal width: 160

  1. x = 8y A) Vertex: (0, 0); Focus: 0, 1 32

; Directrix: y = - 1 32

; Focal width: 32

B) Vertex: (0, 0); Focus: 1 32

, 0 ; Directrix: x = 1 32

; Focal width: 32

C) Vertex: (0, 0); Focus: 1 8

, 0 ; Directrix: x = - 1 8

; Focal width: 0.

D) Vertex: (0, 0); Focus: 1 32

, 0 ; Directrix: x = - 1 32

; Focal width: 0.

  1. y2^ = 8x A) Vertex: (0, 0); Focus: (0, 2); Directrix: y = -2; Focal width: 2 B) Vertex: (0, 0); Focus: (2, 0); Directrix: x = -2; Focal width: 8 C) Vertex: (0, 0); Focus: (2, 0); Directrix: x = -2; Focal width: 32 D) Vertex: (0, 0); Focus: (2, 2); Directrix: x = 2; Focal width: 32

  2. (y - 8)2^ = 16(x - 2) A) Vertex: (2, 8); Focus: (6, 8); Directrix: x = -2; Focal width: 16 B) Vertex: (8, 2); Focus: (8, 18; Directrix: y = -14; Focal width: 16 C) Vertex: (2, 8); Focus: (18, 8); Directrix: x = -14; Focal width: 16 D) Vertex: (8, 2); Focus: (8, 6); Directrix: y = -2; Focal width: 4

  3. (x - 8)2^ = 16(y - 6) A) Vertex: (6, 8); Focus: (10, 8); Directrix: x = 4; Focal width: 4 B) Vertex: (6, 8); Focus: (22, 8); Directrix: x = -8; Focal width: 16 C) Vertex: (8, 6); Focus: (8, 10); Directrix: y = 2; Focal width: 16 D) Vertex: (-8, - 6); Focus: (-8, 10); Directrix: y = -22; Focal width: 16

Find the standard form of the equation of the parabola.

  1. Vertex at the origin, focus at (0, 9) A) y = 1 9

x2^ B) y2^ = 36x C) y2^ = 9x D) y = 1 36

x

PreCalculus

  1. Focus at (0, 4), directrix y = - 4 A) y2^ = 4x B) y = 1 4

x2^ C) y = 1 16

x2^ D) y2^ = 16x

  1. Vertex at the origin, focus at (2, 0) A) y2^ = 8x B) x = 1 8

y2^ C) y = 1 8

x2^ D) x2^ = 8y

  1. Focus at (-2, 0), directrix x = 2 A) x = - 1 8

y2^ B) - 8y = x2^ C) y2^ = - 8x D) y = - 1 8

x

  1. Focus at (-3, 2), directrix x = - 11 A) (y - 2)2^ = 16(x + 3) B) (x + 3)2^ = 16(y - 2) C) (x - 2)2^ = 16(y + 7) D) (y - 2)2^ = 16(x + 7)

  2. Focus at (6, - 2), directrix y = - 8 A) (y + 2)2^ = 12(x - 6) B) (x - 6)2^ = 12(y + 2) C) (x - 6)2^ = 12(y + 5) D) (x + 2)2^ = 12(y + 5)

  3. Vertex at the origin, opens to the right, focal width = 14 A) y2^ = 14x B) y2^ = - 14x C) x2^ = 14y D) y2^ = 3.5x

Graph the parabola.

  1. 8y = x

-10 -5 5 10 x

y 10

5

-10 -5 5 10 x

y 10

5

PreCalculus

  1. y = 4 7

(x + 4)2^ - 3

-10 -5 5 10 x

y 10

5

-10 -5 5 10 x

y 10

5

  1. x = - 3(y + 2)2^ - 1

-10 -5 5 10 x

y 10

5

-10 -5 5 10 x

y 10

5

Find the vertex, the focus, and the directrix of the parabola.

  1. x2^ + 2x - 8y - 39 = 0 A) Vertex: - 1, 1 ; Focus: (-1, 3); Directrix: y = 13 B) Vertex: - 1, - 5 ; Focus: (-1, - 3); Directrix: y = - 7 C) Vertex: - 1, - 4 ; Focus: (-1, - 7); Directrix: y = - 3 D) Vertex: - 1, - 39 8

; Focus: (-1, 3); Directrix: y = - 41 8

  1. 3x2^ - 30x - y + 77 = 0 A) Vertex: 5, 24 83

; Focus: 5, 23 12

; Directrix: x = 25 12

B) Vertex: 5, 2 ; Focus: 5, 25 12

; Directrix: y = 23 12

C) Vertex: 5, 5 ; Focus: (5, 5); Directrix: x = - 1 D) Vertex: 5, 19 2

; Focus: (5, 14); Directrix: x = 10

  1. y2^ - 8x - 6y + 17 = 0 A) Vertex: 5, 1 ; Focus: (9, 1); Directrix: y = 7 B) Vertex: 0, 1 ; Focus: (-1, 1); Directrix: y = 3

C) Vertex: - 4, 3 ; Focus: 98 , 3 ; Directrix: y = 78 D) Vertex: 1, 3 ; Focus: (3, 3); Directrix: x = - 1

Answer Key

Testname: 10_PARABOLAS

1) A

2) A

3) D

4) B

5) A

6) C

7) D

8) C

9) B

10) A

11) D

12) C

13) A

-10 -5 5 10 x

y 10

5

-10 -5 5 10 x

y 10

5

-10 -5 5 10 x

y 10

5

-10 -5 5 10 x

y 10

5

Answer Key

Testname: 10_PARABOLAS

-10 -5 5 10 x

y 10

5

-10 -5 5 10 x

y 10

5

  1. B
  2. B
  3. D