Write the equation for a parabola with the focus at (-1, 4) and the equation of the directrix x = 5.

A standard form of the equation of a rotated parabola is

(y - k) ² = 4p (x - h)

where

(h, k) is the location of the vertex.

(h+p, k) is the location of the focus.

The directrix is the line x = h - p

Because the focus is at (-1, 4), therefore

h + p = - 1 (1)

k = 4 (2)

Because the directrix is x = 5, therefore

h - p = 5 (3)

Add equations (1) and (3) to obtain

h + p + (h - p) = - 1 + 5

2h = 4

h = 2

From (1), obtain

p = - 1 - h = - 1 - 2 = - 3

The equation of the parabola is

(y - 4) ² = - 12 (x - 2)

Answer: (y - 4) ² = - 12 (x - 2)