Final answer:
To derive the equation of the parabola, we need to find the coordinates of the vertex and the equation of the directrix. The equation of the parabola with a focus at (6, 2) and a directrix of y = 1 is f(x) = -1/2(x - 6)^2 + 3.
Step-by-step explanation:
To derive the equation of the parabola, we need to find the coordinates of the vertex and the equation of the directrix.
Since the focus is at (6, 2), the x-coordinate of the vertex is 6. The distance from the vertex to the directrix is equal to the distance from the vertex to the focus.
Therefore, the y-coordinate of the vertex is 1 unit above the focus, which gives us a vertex of (6, 3).
The parabola has a vertical axis of symmetry, so the equation of the parabola can be written as f(x) = a(x - h)^2 + k. Substituting the vertex coordinates into the equation, we have f(x) = a(x - 6)^2 + 3.
Now we can find the value of a by using the distance formula between a point on the parabola and the directrix. Let's take a point (x, y) on the parabola and substitute the equation of the directrix y = 1 into the distance formula.
We get |y - 1| = √((x - 6)^2 + (y - 2)^2). Squaring both sides and simplifying, we have (y - 1)^2 = (x - 6)^2 + (y - 2)^2. Moving all terms to one side, we get 2y^2 - 12y + 16 = 0.
Comparing this with our general equation f(x) = a(x - 6)^2 + 3, we can see that a = -1/2. Therefore, the equation of the parabola is f(x) = -1/2(x - 6)^2 + 3.