CPhill

If AD  is parallel  to  BE....then  angles   ADC  and BEC  are equal

But  angle   ACD  is  greater than angle  BCE

So.. in triangles  CAD  and CBE.......this means that  angle  CAD  is less than angle  CBE

The directrx  is  below the focus.....so this parabola turns upward

p =  the  distance  between  the  focus and directrix  / 2   =   (4 - - 6)  / 2  = 10 /2  = 5

So the  vertex =   ( - 4,  4 - p)   =  ( -4, 4 -5)  = (-4, -1)

So  we  have   this form

4p ( y - k)  =  (x - h)^2            where  (h, k)  is the vertex.......so....we have

4(5) ( y -  -1)   = ( x   + 4)^2      simplify

20 (y + 1)  = ( x + 4)^2

20y + 20   = x^2  + 8x  + 16       subtract  20 from both sides

20y =   x^2  + 8x   - 4      divide  both sides by 20

y = (1/20)x^2 + (2/5)x  - (1/5)

Here's the graph  :  https://www.desmos.com/calculator/5tzekvcuhj

NVM....

This is  analogous to  finding all  the different  sets  consisiting of the 3 defective  phones and any 2  of the other 19 non-defective  phones

So.....the  number  of ways  =  C(19,2)   =  171  ways

The  orthocenter  is  the  intersection point  of  a triangle's altitudes.....we  have....

Slope  of  line  containing side  JL  =  [6-1]/ [ 4 - 2]  =  5/2

Equation of   perpendicular line passing through K :

y = (-2/5)(x - 9)  + 1

y = (-2/5)x + 18/5 + 1

y =  (-2/5)x + 23/5      (1) 

Slope  of line containing  side  KL  =   [ 6 - 1] / [4 - 9]  =  5/-5  = -1

Equation of a perpendicular  line passing  through J

y = 1 ( x - 2)  + 1

y = x - 1      (2)

Equate  (1)  and (2)  to find  the  x coordinate of the orthocenter

(-2/5)x + 23/5  = x  - 1     rearrange as

23/5 + 1  = ( 2/5 + 1)x

28/5  = (7/5)x

28  = 7x

4 =  x

And

y = 4  -1   = 3

So....the orthocenter is  (4,3)

(a)  All hit  the  target =  (2/5)(1/2) (3/4)  = 6 / 40  =  3/20

(b)   ????

(c)  None hit  the target  =  (3/5)(1/2)(1/4)  =  3 / 40

(d)  P(one hits  the  target)  = 

P(first hits  the target  and other two miss)  = (2/5)(  1/2) (1/4)  = 2/40

P(second  hits  the target and first and third miss)  = (3/5) (1/2) (1/4)  = 3/40

P(third hits  the target and first two miss)  = (2/5)(1/2)(3/4)  =  6/40

So  P (one  hits the target  )  =  2/40 + 3/40 + 6/40  =  11 / 40

Number  red  = R

Number  blue  = 3R - 5

So

R  + 3R - 5  = 27     

4R  -  5  = 27    add  5 to both sides

4R  = 32   divide both sides  by   4

R  = 8     

Number blue  =  3(8)  - 5  =   19 

Well  she  can  only shoot  at  ducks 2 through  9 because  there  is no duck to the left  of  duck 1  or to the right of duck 10

So.......the number  of  ways  (orders)  of ducks she  can shoot  is

[ 9 - 2  + 1 ] !   =   8!   =   40320   ways

This is  a  "same degree / same degree"  rational function

The  horizontal asymptote  will  be   y =  the  ratio  of the  coefficients  on  both of the 4th power terms  in the  numerator/ denominator

So  the horizontal asymptote  is

y =  12 / 3    =   4

Se the  graph here to  confirm this  :

https://www.desmos.com/calculator/hjmx53zp3d

Let  C be the  rate of the current

Downstream   the rate of the  boat =  10 + C

Upstream the  rate =  10 - C

So    rate * time  =  distance.....so   equating distances  we have  that

(10+C  ) * 2  =   (10 - C)  * 3       (1)

20  + 2C =  30  - 3C     rearrange  as

5C  =  10           divide both sides  by  5

C = 2     [ the speed of the  current = 2 mph ]

And  using either side of   (1).....the distance is  2 (10 + 2)  = 24  miles