CPhill

The answer  lies  in the  solution to this : https://web2.0calc.com/questions/angle-bisectors_36

The math books can occupy  any of  4 positions

And for each of these positions they can be arranged in 4! = 24 ways

And for each of these the psychology books can be arranged in 3! = 6 ways

So

4 * 24 * 3  =  288 arrangements

                C

            6

       Y              6

    12

A              Z                B    

              12

Because BY is a  bisector

BC / CY  = AY / AB

BC / 6 = 12 /12

BC  = 6

And  because CZ is a  bisector

AZ / BZ  = AC / BC

AZ / BZ = 18 / 6

AZ / BZ  = 3

AZ = 3BZ

So     

BZ + AZ   = AB

BZ + 3BZ = 12

4BZ  =12

BZ =  3

xy  = 2/5

y = 2 / (5x)

6x + 3 / (2 / (5x) )  = 

6x +  (15/2)x  = 

(27/2) x

This has no min

x^2 -14x  + y^2 + 16y + 26

Take the derivative with respect to x and  set to 0

2x - 14   = 0

2x = 14

x = 7

Take the derivative with respect to y and  set  to 0

2y + 16 = 0

2y = -16

y = -8

The min is    (7)^2  - 14(7) + (-8)^2 + 16(-8) + 26  =  -87

OP^2         =       AP^2

x^2 + y^2   = (x + 2)^2  + (y -1)^2

x^2 + y^2 = x^2 + 4x + 4  + y^2  - 2y + 1

2y - 1  = 4x + 4

2y    = 4x + 5

y = 2x + 5/2         (1)

OP^2  =    BP^2

x^2 + y*2  =   (x + 3)^2  + (y - 2)^2

x^2 + y^2   =  x^2 + 6x + 9  + y^2  - 4y + 4

4y - 4 = 6x + 9    (2)

Sub (1)  into (2)

4 (2x + 5/2) - 4 =  6x + 9

8x + 6   = 6x + 9

2x = 3

x = 3/2

y  = 2(3/2) + 5/2   =   11/2

Coordinates of center =  (3/2 , 11/2)

See a similar problem here : https://web2.0calc.com/questions/parabola_2038

(r-10)^2 + (r -13)^2  = r^2

r^2 - 20r + 100 + r^2 -26r + 169  = r^2

r^2 - 46r + 269  =  0

r = 23 + 2sqrt (65)  ≈  39.125

\(\frac{AC}{BC}\)

Not sure what part the semi-circle  plays

angle ABC  =60

angle BAC = 30

AC / BC  =  sqrt (3)  /  1

Let  QR  =  4 + x

Let the radius of the larger circle be  PM

Let the radius of the smaller circle br QN

We can  construct similar triangles PMR  and  QNR  such that

PR / PM  = QR / QN

( 13 + x)  / 5 =  (4 + x)  / 4

4 (13 + x) =  5 (4 + x)

52 + 4x   = 20 + 5x

x = 32

QR =  4 + 32   =  36