I've answered the first one elsewhere today........
Geometrically speaking, a parabola is defined as the set of points that are the same distance from a given point and a given line. The point is called the focus of the parabola and the line is called the directrix of the parabola. Suppose P is a parabola with focus (4,3) and directrix y=1 . The point (8,6) is on P because (8,6) is 5 units away from both the focus and the directrix. If we write the equation whose graph is P in the form y=ax^2 + bx + c, then what is a*b*c ?
The vertex will be found at (4,2) and we can write that :
(y - 2) = (a)(x - 4)^2 and since (8,6) is on the curve, we can solve for a
(6 -2) = (a) (8 - 4)^2 simplify
4 = (a)(4)^2
4 = (a)16 → a = 4/16 = 1/4
Since the x coordinate of the vertex is given by -b/ (2a) we have that -b/[2 (1/4)] = 4 → -b / (1/2) = 4 → -b = 2 → b = -2
And using the fact that (4,2) is on the graph, we can find c, thusly :
y = ax^2 + bx + c ......so......
2 = (1/4)(4)^2 -2(4) + c
2 = 4 - 8 + c
2 = -4 + c
c = 6
Then a*b*c = (1/4) (-2) (6) = (1/4)(-12) = -3
+1094 
