Answer:
x = 3/2 - ((y + 4)²)/2
Explanation:
The coordinates of the focus of the parabola = (1, -4)
The directrix of the parabola is the line, x = 2 (parallel to the y-axis)
The general form of a parabola having a directrix parallel to the y-axis is presented as follows;
(y - k)² = 4·p·(x - h)
The coordinates of the focus = (h + p, k)
The line representing the directrix is x = h - p
Therefore, by comparison, we have;
(1, -4) = (h + p, k)
k = -4
h + p = 1
∴ p = 1 - h
From the directrix, we have;
2 = h - p
∴ 2 = h - (1 - h) = 2·h - 1
2·h = 2 + 1
h = 3/2
p = 1 - h
∴ p = 1 - 3/2 = -1/2
p = -1/2
By substitution of the values of k, p, and h, in the general equation of the parabola, (y - k)² = 4·p·(x - h), we get;
(y - k)² = 4·p·(x - h) = (y - (-4))² = 4·(-1/2)·(x - (3/2)) = (y + 4)² = -2·(x - 3/2)
The equation of the parabola in vertex form is therefore;
x = (-1/2)·(y + 4)² + 3/2 = 3/2 - (y + 4)²/2