Final answer:
To find the equation of the parabola with focus (-5,4) and directrix x = -1, we use the distance between the focus and directrix to determine the vertex and then write the equation in vertex form as (x + 7)^2 = 8(y - 4).
Step-by-step explanation:
The equation of a parabola can be derived using the definition that each point on the parabola is equidistant from the focus and the directrix. Let's find the equation of a parabola with focus (-5,4) and directrix x = -1.
Steps to Write the Equation of the Parabola:
- Determine the distance between the focus and the directrix. This distance is 4 units, since the focus is at x = -5 and the directrix is at x = -1. So the vertex of the parabola will be 2 units to the left of the focus, at x = -7.
- Knowing the vertex (h, k) is now (-7, 4), we can write the equation in vertex form given by (x-h)^2 = 4p(y-k), where p is the distance from the vertex to the focus, which is 2 in this case.
- Substitute the values into the vertex form equation and get (x + 7)^2 = 8(y - 4), which represents the parabola with the given focus and directrix.
Therefore, the equation of the parabola is (x + 7)^2 = 8(y - 4).