Final answer:
The equation for the parabola with the given focus and directrix is ((y-5)^2 = 42(x+3.5).
Step-by-step explanation:
The equation for a parabola with a focus at (0,5) and a directrix at x=-7 can be derived using the definition of a parabola. A parabola is the set of all points that are equidistant from a single point, called the focus, and a line, called the directrix. We'll use the general form of a parabola's equation, which for a vertical axis of symmetry is (x-h)^2 = 4p(y-k), where (h, k) is the vertex of the parabola and p is the distance from the vertex to the focus (if the parabola opens upwards) or the distance from the vertex to the directrix (if the parabola opens downwards).
In this case, since the directrix is a vertical line x=-7, we know that the axis of symmetry of the parabola will be horizontal, and the parabola will open rightwards since the focus is to the right of the directrix. The vertex lies halfway between the directrix and the focus, so its x-coordinate is halfway between -7 and 0, which is -3.5. The y-coordinate of the vertex will be the same as the focus, so it is 5. The distance p will be half of the distance between the focus and the directrix, hence p = 7 - (-3.5) = 10.5. We now place these values into the general form of the equation for a horizontal parabola (y-k)^2 = 4p(x-h), and we get:
(y-5)^2 = 42(x+3.5)