Find the equation of the parabola with it's focus at (-4,7) and directrix y=1

Is y+4=-1/12(x-4)^2 Right or is it
y-7=1/24(x=a)^2
y-4=1/12(x+4)^2
y-3=1/24(x+4)^2

The vertex is between the directrix nd the focus, at y = 4. It is also at

x = 4, which means the equation must contain an x-4 term. That leaves you with
y-4 = 12(x+4)^2

To find the equation of a parabola given its focus and directrix, we need to use the standard form of the equation for a parabola: (x - h)^2 = 4a(y - k), where (h, k) is the vertex and "a" is the distance between the vertex and the focus (or vertex and the directrix).

Given that the focus is (-4, 7) and the directrix is y = 1, we can determine the vertex and the value of "a".

The vertex is equidistant from the focus and the directrix, so the y-coordinate of the vertex is the average of the y-coordinate of the focus and the y-coordinate of the directrix.

Vertex: y = (7 + 1) / 2 = 4

The distance between the vertex and the focus (or directrix) is equal to "a". So, a = 7 - 4 = 3.

Now, we have the vertex (h, k) = (0, 4) and "a" = 3.

Plugging these values into the standard form equation, we get:

(x - 0)^2 = 4 * 3(y - 4)
x^2 = 12(y - 4)
x^2 = 12y - 48

Therefore, the equation of the parabola with focus (-4, 7) and directrix y = 1 is x^2 = 12y - 48.

To find the equation of a parabola given its focus and directrix, we can use the standard form of the equation of a parabola:

For a parabola with vertical axis of symmetry:
(y - k)^2 = 4p(x - h)

Where the vertex is given by (h, k), the focus is (h, k + p), and the directrix is a horizontal line given by y = k - p.

In this case, the focus is (-4,7) and the directrix is y = 1.

1. Determine the vertex:
The vertex is halfway between the focus and directrix along the axis of symmetry. In this case, the axis of symmetry is vertical, so the x-coordinate of the vertex is -4. The y-coordinate can be found by averaging the y-coordinates of the focus and the directrix: (7 + 1) / 2 = 8 / 2 = 4. Therefore, the vertex is (-4, 4).

2. Determine the value of p:
The distance between the vertex and the focus (or vertex and the directrix) is denoted by the letter p. In this case, the distance between the vertex (-4, 4) and the focus (-4, 7) is 7 - 4 = 3. So p = 3.

3. Write the equation:
Using the values we have obtained, we can write the equation of the parabola:

(y - 4)^2 = 4 * 3 * (x - (-4))

Simplifying this equation gives:

(y - 4)^2 = 12(x + 4)

So, the correct equation for the parabola is y-4=1/12(x+4)^2.

Therefore, among the options given, y-4=1/12(x+4)^2 is the correct equation.