M6 HW Worksheet 9

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M6 HW Worksheet 8. 02/13/2012. Sussman. Find the ... ⋅2. 1. 2. ⋅2. = 5. 3. Major axis length 2⋅a = 2⋅4 = 8. Major axis length 2⋅a = 2⋅. 1. 2. =1. Minor axis ...
M6  HW  Worksheet  8     02/13/2012       Sussman     Find  the  vertices  foci,  and  eccentricity  of  the  ellipse.   Determine  the  lengths  of  the  major  and  minor  axes,  and  sketch  it  graph.     1.   x 2 + 4 y 2 = 16       2.   9x 2 + 4 y 2 = 1                       1 2 1 2 9x 4y 9 4 + =1 2 2 x y 1 1 + =1 16 4 9 4     2 2 x2 y2 x y + =1 + =1 2 2 42 22 ⎛ 1⎞ ⎛ 1⎞ ⎜⎝ 3 ⎟⎠ ⎜⎝ 4 ⎟⎠   c2 = c 2 = a 2 − b2 c= a −b 2

Foci  

2

c = 16 − 4

(−2 3,0),(2 3,0)  

c = 12 = 2 3

9 4 − 36 36 ⎛ 5⎞ ⎛ 5⎞ Foci 0,− , 0, ⎜ ⎟ ⎜ ⎟   6 ⎠ ⎝ 6 ⎠ 5 ⎝ c= 36 c=

c=

2 3 3 = Eccentricity.     4 2 Major  axis  length   2 ⋅ a = 2 ⋅ 4 = 8  

1 1 − 4 9

5 6

5 5 ⋅2 5 Eccentricity.   6 = 6   = 1 1 3 ⋅2 2 2 1 Major  axis  length   2 ⋅ a = 2 ⋅ = 1   2

  Minor  axis  length   2 ⋅ b = 2 ⋅ 2 = 4      

1 2 Minor  axis  length   2 ⋅ b = 2 ⋅ =   3 3

1 1 1 3..   x 2 + y 2 =   2 8 4   4 2 4 2 x + y =1 2 8 1 2x 2 + y 2 = 1 2 1 2 1 2x 2 y 2 2   + 2 =1 1 2 2 2 x y2 + =1 1 2 2 x2 y2 + =1 2 2 ⎛ ⎛ 1 ⎞ 1/2 ⎞ 2 ⎜⎜ ⎟ ⎟ ⎝ ⎝ 2⎠ ⎠

   

 

( )

 

 

c = a −b 1 4 1 3 Foci   c 2 = 2 − = − =   2 2 2 2 2

c=

2

3

2

=

6 2

2 ⎛ 6⎞ ⎛ 6⎞ ⎜ 0,− ⎟ , ⎜ 0, ⎟   2 ⎠ ⎝ 2 ⎠ ⎝

6 6 3 Eccentricity   2 =   = 2 2 2 2 Major  axis   2a = 2 ⋅ 2 =   Minor  axis   2b = 2      

2 = 2   2

Find  an  equation  for  the  ellipse     whose  graph  is  shown.            

 

 

 

 

 

 

 

 

                   

c 2 = a 2 − b2 21.  

4 = a2 + 4 a2 = 8

x2 y2 + = 1   2 8

a=2 2    

c 2 = a 2 − b2 32 = 42 − b2 x 2 y 2 + = 1   22.   2 7 16 b =7 b= 7      

19.

x2 y2 + = 1   52 42

20.

x2 y2 + = 1   22 52