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Please help, ty! Apr 23, 2018

#1
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4x^2 + 4y^2 - 4x + 12y - 6  = 0     add 6 to both sides

4x^2  - 4x + 4y^2 + 12y   =  6        complete the square on x and y

4(x^2 - x + 1/4)  + 4 (y^2 + 3y + 9/4)   =  6  + 1  + 9

4(x - 1/2)^2  +  4(y + 3/2)^2  =  16       divide through by 4

(x - 1/2)^2  + ( y + 3/2)^2  =   4   ⇒   "D"   Apr 23, 2018
#2
+2

center (-2,3)   point  (4, -3)

Since  (4, - 3)  is on the circle, we need to find the distance  between this point and the center....that will be the radius, r...so we have...using the distance formula

√ [ (-2-4)^2  + ( -3 - 3)^2  ]  =  √ [ (-6)^2 + (-6)^2]  = √ [36 + 36]  = √72   = r

So....the equation is

(x  - h)^2  + (y - k)^2  = r^2      where (h, k)  is the center and  r^2  = 72

(  x -  -2)^2 + ( y - 3)^2  = 72

(x + 2)^2  + ( y - 3)^2  = 72 ⇒    "A"   Apr 23, 2018
#3
+2

focus   ( 4, - 3)    directrix   x = -2

Since the directrix lies to the left of the focus, this parabola  opens to the right

The y coordinate of the vertex will  be  -3

The x coordinate will be :  [x coordinate of the focus plus the directrix] / 2  =     [ 4  +  - 2] / 2   =   2/2  = 1

We have this form

4p ( x - h)  =  (y - k)^2     where  the vertex  is  (h, k)  = ( 1 , -3)

And  p  will be the distance between the vertex and the focus  =  3 units....since the parabola opens to  the right, this will be  positive

So....putting this all together, we have

4(3)(x - 1)  = ( y -  -3)^2

12 ( x - 1)  =  ( y + 3)^2

(y + 3)^2  = 12 ( x  - 1)   ⇒  "D"   Apr 23, 2018