Directions: The focus and directrix of a parabola are given. Write an equation for each parabola.

(2,4) y = 6

May you explain the procedures on how to solve this equation? Please be detailed on each step.

I assume you have to use the basic definition of a parabola as the set of points equidistant from the focal point (2,4) and the line y=6

Make a sketch and let P(x,y) be any point on the parabola

so √((x-2)^2 + (y-4)^2) = √(y-6)^2

square both sides
(x-2)^2 + (y-4)^2 = (y-6)^2

x^2 - 4x + 4 + y^2 - 8y + 16 = y^2 - 12y + 36

y = -(1/4)x^2 + x + 4

To write the equation of a parabola given its focus and directrix, we need to use the standard form of the equation for a parabola. The standard form is as follows:

(y - k)^2 = 4p(x - h)

where (h, k) represents the coordinates of the vertex and p represents the distance between the vertex and the focus (or the vertex and directrix).

In this case, the given directrix is y = 6. To find the value of p, we need to determine the perpendicular distance from the vertex to the directrix. Since the directrix is a horizontal line with y = 6, the perpendicular distance is the difference between the y-coordinate of the focus (4) and the directrix (6). Therefore, p = 4 - 6 = -2.

Now, we have the coordinates of the vertex (2,4) and the value of p (-2). We can substitute these values into the standard form equation and simplify to obtain the equation of the parabola:

(y - 4)^2 = 4(-2)(x - 2)

(y - 4)^2 = -8(x - 2)

This is the equation of the parabola with the given focus (2,4) and directrix y = 6.