Final answer:
To derive the equation of the parabola with the given focus and directrix, find the vertex as the midpoint, calculate the distance to the focus (p), and plug these values into the parabolic equation (y - k)^2 = 4p(x - h). The resulting equation is (y - 3)^2 = 10(x - 3.5).
Step-by-step explanation:
To find the equation of a parabola given a focus at (6,3) and the directrix at 4x = 4, first simplify the directrix equation to x = 1. The formula for a parabola that opens either left or right is (y - k)^2 = 4p(x - h), where (h, k) is the vertex of the parabola, and p is the distance from the vertex to the focus or directrix.
Since the focus has a y-coordinate of 3, the axis of symmetry of our parabola is y = 3. The x-coordinate of our vertex will be midway between the directrix and the focus, which is at (3.5,3). Thus, p = 2.5 since it is the distance from the vertex to the focus at (6,3), and also equal to the distance from the vertex to the directrix x = 1. We can now substitute the values into the equation: (y - 3)^2 = 4(2.5)(x - 3.5), simplifying to (y - 3)^2 = 10(x - 3.5).