+129852 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"
+129852 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"
+129852 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"
