inal answer:
The equation of the parabola with focus (3, -5) and directrix y = 5 is y = -1/20(x - 3)².
Step-by-step explanation:
To find the equation of a parabola with a focus at (3, -5) and a horizontal directrix at y = 5, we must consider the definition of a parabola. A parabola is the set of all points equidistant from the focus point and the directrix. The vertex of the parabola will be midway between the focus and directrix. Given that the directrix is at y = 5 and the focus is at (-5), the vertex will be at y = 0 (as it's the midway point between -5 and 5). Now we know the vertex form of a parabola is y = a(x - h)² + k, where (h, k) is the vertex.
Since the focus is below the directrix, the parabola opens downwards and thus 'a' will be negative. The vertex, in this case, will be (3, 0). The distance from the vertex to the focus or to the directrix is 5 units, which is also the absolute value of the focal length (|p|). As the parabola opens downwards, a will be -1/(4p), so a is -1/20. The final equation comes out to be y = -1/20(x - 3)².