Write the equation for a parabola with a focus at (1,-4)(1,−4)left parenthesis, 1, comma, minus, 4, right parenthesis and a directrix at x=2x=2x, equals, 2.

x = 3/2 - ((y + 4)²)/2

Step-by-step 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