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Answer:
- p = -1
- focus: (-1, 3)
- directrix: y = 5
Step-by-step explanation:
The latus rectum is the horizontal chord through the focus. It intersects the parabola at a point that is equidistant from the focus and the directrix (as all points on the parabola are). The vertex is half the distance from the focus to the directrix, so the points of intersection are twice as far horizontally from the focus as they are vertically from the vertex.
The short of it is this: draw a line with slope 1/2 through the vertex. Where it intersects the parabola is one end of the latus rectum. The y-coordinate of the point will be the y-coordinate of the focus.
Here, the vertex is (-1, 4). A line of slope 1/2 through that point will intersect the parabola at (-3, 3). This means the focus is (-1, 3), one unit below the vertex. That -1 unit is the value of p. The location of the directrix is that 1 unit in the opposite direction from the vertex, at y = 5.
