CPhill

Rearrange the given equation as

x^2 - 6x + y^2 + 8y  = 24     complete the square on x and y

x^2 - 6x + 9  + y^2 + 8y + 16  =  24  + 9 + 16

(x -3)^2  + (y + 4)^2  =  49

The center of the circle  is  ( 3, -4)

So.....the distance between the center  and (-3, -12)   is given by :

sqrt [ (-3 - 3)^2  + (-12 - - 4)^2 ] =

sqrt  [ (-6)^2  + (-8)^2 ] =

sqrt [ 36 + 64 ] =

sqrt (100) =

10 units

We can solve this by inspection. Logic

If  abc  =  21     and    a + 1/b  =  22/7

Then this implies that

a + 1/b  =  3  + 1/7

Which implies that  a  = 3    and  b  = 7

So

abc  =21

(3)(7)(1)  = 21

So  c  = 1

And

c + 1/a  =    1   + 1/3

See my answer here, Logic  :  https://web2.0calc.com/questions/can-someone-help-me-with-this_4

The arthmetic "formula"   for  the number of pages read on Day  N  is given by :

5  + (N-1)(.5)  =

5 + .5N - .5 =

4.5 + .5N

The first term  =  5

And the last term is    4.5 + .5N

So....we want to solve this :

(N/2)  [ first term + last term ] = 80

(N/2)[ 5 + 4.5 + .5N ]  = 80

(N/2) [ 9.5 + .5N ]  = 80       mutiply through by 2

N [ 9.5 + .5N ]  =  160

.5N^2 + 9.5N  - 160  = 0        multiply through by 2 again

N^2  + 19N - 320  = 0

The graph here (letting N = x)  shows two possible solutions  :  https://www.desmos.com/calculator/ze8gobvp4n

Taking the positive one...it will take about 10.75 days  =  11  days

Hannah starts at  (-4, 9)    walks 10 units to the left and then 7 units down

So.....she ends up at   ( -4 -10 , 9 - 7)   =  (-14, 2)

So....we need to find the slope  between    (-4,9)   and (  -14, 2)

We have

[ 2 - 9 ] / [ -14 - - 4]  =    [-7]/ [-10]  =7/10

Using either of the points to write an equation of the line we have that

y  =(7/10)(x - -4) + 9       simplify

y = (7/10)(x + 4) + 9      mutiply through by 10

10y = 7(x + 4) + 90

10y = 7x +28  + 90

10y  =  7x + 118 subtract 7x from both sides

-7x + 10y  = 118

Let the center of the hyperbola be  at the origin  =  (0,0)

We will have this form

x^2           y^2

___    -    ____    =  1

a^2           b^2

c  = the distance from either of the focal points to the center   =  100  miles

And c^2  = 10000

For any point on the hyperbola, the differences of the distances from the two foci  to this point  = 2a

So

2a  = 180

a  =  90    ⇒   a^2  = 8100

And we have that

c^2  = a^2 + b^2

c^2 - a^2  = b^2

10000 - 8100  = b^2

1900  =  b^2

So  the equation becomes

x^2           y^2

____  -    _____   =      1

8100         1900

Here is a graph : https://www.desmos.com/calculator/5dyujkabt4

2)

 12   2                 

 4    -2                 -24 - 8

______   =        _______  =     - 8

  1   2                   -2 +  6

 -3 - 2

1  12          

-3  4               4 + 36

_____  =       ______  =    10

1   2               -2 + 6

-3 -2

1)  We need to find  the inverse matrix  of

3  2

-1 5

This is given by

        1                5    -2      =        1        5   - 2    this goes in the first answer  box

__________      1     3              ___      1     3

3*5 - (2) (-1)                               17

  1               this goes in the second answer box

-23

   1              5   - 2            this goes in the third answer box

_____         1     3

 17

1            5     -2                1

___        1     3        x       -23      =

17

1         5(1) + (-2)(-23)

__       1(1)  + (3)(-23)

17

  1           51           this goes in the  fourth answer box

___        -68

 17

 3                     this goes in the last answer box

-4

2y  =  f(2x) / 2  + 2       divide through by 2

y  = (1/4) f(2x) + 1

The point   (9/2, (1/4)(7) + 1)  =  (9/2, 7/4 + 1)  = (9/2, 11/4)  will be on the graph

The sum of the coordinates  is     9/2 + 11/4  =   18/4 + 11/4  =   29/4

We have these two equations

3  = (2)^2 + 2b + c

3 =  (4)^2 + 4b + c   simplify

3 = 4 + 2b + c

3  = 16 + 4b + c

-1  = 2b + c    ⇒   2 = -4b  - 2c    (1)

-13 = 4b + c      (2)

Add (1) and (2)  and we have

-11 = -c

c  = 11