CPhill

Thanks, tertre     [ and tanmai ]   !!!  

Here's my best attempt, tanmai....

o       o         o         o

o      o         o          o

o      o         o          o 

o      o         o          o   

We have   C(16, 4)   = 1820   possible sets of any four of the points

But.....the only ones that will be collinear points  are the  four horizontal and vertical patterns and the two diagonal patterns   =   10 total

So....the probability   is      10 / 1820   =   1 / 182

We can write the function as

f(x)   =  36 ( x - 1)^(-1)

The derivative is    -36 ( x - 1)^(-2).....set this = - 9

-36(x - 1)^(-2)   = -9

36 (x - 1)^(-2) = 9        rearrange as

36/9  =  ( x - 1)^2

4 = ( x - 1)^2

The x's that solve this   are   x =  3  and x = -2 

When x = 3....y = 36 / [ 3 - 1]   =   18

When x = -2....y =  36 / [ -2 - 1]  =  -12

So....the two points of tangency are   ( 3, 18)   and (-2, -12)

Here is the graph : https://www.desmos.com/calculator/pzmmsxfr2z

OK.......good deal.....!!!

Angle CAD is an inscribed angle intersecting arc CD.....so..its measure is 1/2 of the arc = 30°

And since angle ABO = 60°, then angle BOA  =  180 - 30 - 60   = 90°

So....we have 30 - 60 - 90 right triangle

And the side opposite the right angle = AB  =  2BO   = 10

And the side opposite the  the 60° angle ABO   =  AO =  5√3   = √75 =   the radius of the circle

Draw OB through to intersect the circle at M.....so  OM is a radius = 5√3 = √75

So BM = OM - BO =   [ 5√3 - 5]   = [ √75 - 5 ]

And draw BO through to intersect the circle at N

So.....BN =  [ 5√3 + 5] = [ √5 + 5 ]

An

So.....we can use the intersecting chord theorem to find BC

AB * BC   =  BM * BN

10 * BC =  [ √75 - 5 ]  * [ √75 + 5 ]

10 * BC = 75 - 25

10 * BC  = 50     divide both sides by 10

BC =  50 / 10    =   5

Let the center be (x, 0)

The points will be equidistant from the center..so....we can solve this

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

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

6x = 4x   ⇒   x = 0

So.....the center is (0, 0)

And the radius   is     sqrt ( (-3 - 0)^2 + ( 2 - 0)^2 )  =  sqrt [ 9 + 4 ]   = sqrt (13)

Since we can position the vectors as we wish

Let both have their initial point at (0,0)

Let the terminal point of u   = (2,0)

And let the terminal point of v be   ( 7cos (40°)  ,  7sin (40°)  )

The x component  of the resultant is     2 + 7cos(40°)

The y component of the reultant is  7sin (40°)

The magnitude of the resultant is

√ ( ( 7cos (40°) + 2 )^2 +   (7sin (40°))^2 ) ≈ 8.6 units 

f(x)   = cos x / sinx   =  cot x

The derivative of  cot x   =  -csc^2 x

The derivative of the function is

3x^2 - 4x....set this equal to  - 1   and solve for x

3x^2 - 4x  = -1    

3x^2 - 4x + 1    =    0

(3x - 1) ( x - 1)   = 0

Setting each factor to 0 and solving for x produces

 x = 1/3       or         x  = 1

And when x = 1/3, y = -32/27

And when x = 1, y = -2

So.....we have two points where this is true

(1/3, -32/27)    and ( 1, - 2 )

Thanks, tertre....!!!