derive the equation of the parabola for which y=1 is the directrix and F(3,-2) is the focus

Sure, let's derive the equation of the parabola step by step.

The general equation of a parabola is given by (x-h)² = 4a(y-k), where (h,k) is the vertex of the parabola and 'a' is the distance from the vertex to the focus or from the vertex to the directrix.

1. The directrix of the parabola is y = 1 and the focus is F(3,-2). The vertex of the parabola is always halfway between the focus and the directrix. So, the y-coordinate of the vertex is the average of the y-coordinate of the focus and the directrix, which is (-2+1)/2 = -0.5. The x-coordinate of the vertex is the same as the x-coordinate of the focus, which is 3. So, the vertex (h,k) of the parabola is (3,-0.5).

2. The value of 'a' is the distance from the vertex to the focus or from the vertex to the directrix. So, a = |-0.5 - (-2)| = 1.5.

3. Substituting h = 3, k = -0.5, and a = 1.5 into the general equation of a parabola, we get (x-3)² = 4*1.5(y - (-0.5)).

4. Simplifying this equation, we get (x-3)² = 6(y + 0.5).

5. Further simplifying, we get (x-3)² = 6y + 3.

6. Finally, subtracting 3 from both sides, we get (x-3)² - 3 = 6y.

So, the equation of the parabola for which y=1 is the directrix and F(3,-2) is the focus is (x-3)² - 3 = 6y.