CPhill

Factors  of 2008   = 

1  2  4   8     251   502    1004    2008

The smallest  value  for  b  is

( x + 8)  ( x +251)   =    x^2  +  ( 251 + 8) x  + 2008   =   259

THX, guest.....my previous "solution"  was totally wrong !!!

The  vertical asymptotes  will  occur at the  x values   produced by setting the denominator = 0   :

x^2  + 6x   - 7     =  0    factor

( x - 7) ( x + 1) = 0     set each factor to 0  and solve for x

x  -7   =  0                x  +  1   =   0

x  = 7                      x  = -1

These are the equations of  the vertical asymptotes

We have that

1/3  =  7/9  +  d (13 - 1)

1/3  = 7/9  +  12d

(1/3 - 7/9)  =12 d

(3/9 - 7/9)  = 12 d

-4/9  = 12 d      divide both sides  by  12

-4/ 9  *  1/12   = d

-4/108  =d

-1/27    = d

To find the seventh term we have

7/9  +  (-1/27) (7 -1)

7/9 + (-1/27) (6)

7/9  -6/27  = 

7/9 - 2/9  = 

   5/9

If the center lies on the x axis, call the   center  (a, 0)

So.....the distance from this point to each of the given points  is equal

So....equating distances, we have that

(-3-a)^2  + ( 2 -0)^2  = (-2-a)^2 + (3-0)^2        simplify

((-1) (3 + a))^2  + 2^2  =  ((-1)(2 + a))^2  + 3^2

(3 + a)^2  + 4  = (2 + a)^2  +  9

9+ 6a + a^2  + 4  = 4 + 4a + a^2 + 9        subtract a^2  from both sides and simplify

6a +13 = 4a + 13     subtract 13 from both sides

6a  = 4a

This is true when a  = 0

So  the center is    (a, 0)   = (0, 0)

And the radius  is  sqrt  [ ( 0 - -3)^2 +  (0 -2)^2 ] =   sqrt  [ 3^2 +2^2]  = sqrt  ( 13)

Here's a graph :   https://www.desmos.com/calculator/dd6nbqzagb

This cannot be a rectangle....see the graph :

For this to be a rectangle, angle ABC would have to be  = 90°

*****

Nice,heureka  !!!!