CPhill

Tangent -  Secant Theorem

AP^2  = PB( PB + CB)

AP^2  = 4 ( 4 + 12) = 4*16  = 64

AP   =   sqrt (64)   =  8

No solution 

Can't include  6,5,0,7 

Looking at 5201.....either 1 or 2 is  in the number

Can't be 1 because it  isn't in 9285

So it can't include  6,5,0,7,1

So 2 must be one  of the  digits

But it must well-placed in 5201   (in the 2nd  position)  since  5,0,1 aren't in the number     (1)

But since there's only one correct digit in  9285 .it must be 2...but it isn't well-placed (in the second  position) (2)

So....we have a contradiction between (1) and (2)

Can't include 6,4,3 or 1             

8  , 0.....  one is  correct            

Two digits are 2, 7 

8 comes first

8927 

  1 /  [ sqrt 36 - sqrt 27 ] - 1 / [ sqrt 27 - sqrt 18 ] +  1 / [ sqrt 18 - sqrt 9]

1/ [ 6 - sqrt 27 ]  - 1 / [ sqrt 27 - sqrt 18] + 1 / [sqrt18  - 3]

[ 6 + sqrt 27 ] / [ 36 - 27]   - [ sqrt 27 + sqrt 18]  / [ 27 - 18]  + [sqrt 18 + 3 ] / [18 - 9]

[ 6 +sqrt 27 ] / 9   -  [ sqrt 27 + sqrt 18 ] / 9  +  [ sqrt 18   + 3  ]  / 9  

[  6 + 3 ]  / 9  =   

1

\(\frac{xy}{x + y} = 0, \quad \frac{xz}{x + z} = 0, \quad \frac{yz}{y + z} = 1 \)

Either  x  =  0      or    y, z  = 0

But the second cannot be true because it would make the  last fraction undefined

So  x =  0

And  (assuming integer solutions for y, z )

(yz)  /  (y + z)  =  1

yz  =  y + z 

zy  - z  =  y

z ( y - 1)  =  y

z  =  y / (y -1)

y       z

2      2

z =  2

Simplify as

x^2 + (k - 9)x + 16

Note that

(x + 4) (x + 4) =  x^2 + 8x + 16  = (x + 4)^2

So

k -9  = 8

k =17

Also 

(x - 4) (x -4)  = x^2 - 8x + 16  = (x - 4)^2

So

k- 9 = -8

k = 1

17 +  1    =   18

OC  =1

OD  = OC - CD =  1 -1/5 = 4/5 

BD = sqrt [ 1 - (4/5)^2 ] =  sqrt [ 9/25 ]  = 3/5 

ED = (3/5) (1/2) =  3/10

AD = 1 + 4/5 = 1 + 8/10  = 18/10

Triangle ADE is  right with angle ADE =  90......so....using the P Theorem .....

AE = sqrt [ AD^2 +ED^2 ]  = sqrt [ (18/10)^2  + (3/10)^2 ]  = sqrt [ 333 ] / 10 ≈  1.825

a^2 / b  +  b^2 / a  =     [ a^3 + b^3 ] / [ab ] ^2  =   [(a + b)  (a^2  + b^2  - ab ) ] /  [ab]

x^2 -25x + 80  = -28x +  75       simplify

x^2 + 3x + 5 =  0

We have the form  mx^2 + nx + c =  0

Sum of roots

a + b =  -n/m =  -3/1 =  -3

Product of  roots  =  ab =  c / m =  5/1 =  5

a + b =   -3       square both sides

a^2 + 2ab + b^2 =  9

a^2  +2(5) + b^2  = 9

a^2 + 10 + b^2 = 9

a^2 + b^2  =  -1

So

[(a + b)  (a^2 + b^2 - ab) ] /  [ab]  =

[ ( -3)( -1 - 5) ]   / [5] =

[ 18 ] / [5 ]   =  

18 / 5 =  3.6

\($\frac{1}{a+7}+\frac{1}{b+7}.$ \)

Simplify  as

3x^2 -11x -7  =  0

We have the form mx^2 + nx + c  = 0

m =3 , n = -11   c = -7

Product of  roots   = ab = c/m   =  -7/3

Sum of  roots =  

a + b = -n/m  =   11/3          

1/ (a + 7)  +  1 /(b + 7)   =

[ a + 7 + b + 7 ] / [ (a + 7) (b + 7) ]  =

[ a + b + 14 ] / [ ab + 7 (a + b) + 49 ]  = 

[ 11/3 + 14 ]  / [ -7/3 + 7(11/3) + 49 ]  =

[ 53/3 ] / [ 217 /3 ]  =

53 / 217

Nice, EP   !!!!