You have a parabola with the following characteristics:
A. focus (0,1) and directrix x=3.
B. vertex (2,4) and focus (0,4), vertex is (2,4) so h = 2 and k = 4; focus (h+p,k) = (0,4)
In order to determine what is the parabola with the previous conditions, you take into account the following points:
- general form of a horizontal parabola: (y - y0)² = 2p(x - x0)²
- directrix: axis perpendicular to the symmetry axis of the parabola:
x = x0 - p/2
- simmetry axis is at the same point of focus
- focus of the parabola: F(x0 + p/2, y0 )
- vertex: (x0 , y0)
A.
focus F(0,1)
y0 = 1
x0 + p/2 = 0 => x0 = -p/2
directrix x = 3
x = x0 - p/2
3 = x0 - p/2 = x0 + x0 = 2x0
x0 = 3/2
p = -2x0 = -2(3/2) = -3
p = -3
Hence, the equation of the parabola is:
(y - 1)² = -3(x - 3/2)²