Answer:
y = -1/24(x -2)² +6
Explanation:
You want the equation of the parabola with focus (2, -3) and directrix y = 9.
Vertex
The vertex is halfway between the focus and directrix, so will have a y-coordinate of (9 +(-3))/2 = 3. Its x-coordinate is the same as that of the focus.
vertex = (2, 3)
Scale factor
The scale factor of the parabola is 1/(4p), where p is the vertical distance from the focus to the vertex.
p = focus - vertex = -3 -(3) = -6
Then the scale factor is 1/(4p) = 1/(4(-6)) = -1/24.
Equation
The equation of the parabola is ...
y = 1/(4p)(x -h)² +k . . . . . . . for vertex (h, k)
y = -1/24(x -2)² +3
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Additional comment
One end of the latus rectum is the point (14, -3). You will note that it is 12 units from the focus and the same distance from the directrix, as it should be. This serves to check that the equation of the parabola is correct.
As you know, the latus rectum is the segment parallel to the directrix passing through the focus with endpoints on the parabola.
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