A parabola has a focus (5,7) and directrix y =0. Write an equation that says point (x,y) is on the parabola. put it into vertex form, show your work
To write the equation of a parabola in vertex form, we need to use the formula:
(x-h)^2 = 4p(y-k)
where (h,k) is the vertex of the parabola and p is the distance between the focus and the vertex, which is also the distance between the directrix and the vertex.
Given that the focus is at (5,7) and the directrix is y = 0, we can determine that the vertex is halfway between the focus and the directrix. The y-coordinate of the vertex is given by (7+0)/2 = 7/2 = 3.5. Since the directrix is below the focus, the parabola opens upwards, so p = 3.5 - 0 = 3.5.
The vertex is at (5, 3.5), so the equation of the parabola in vertex form is:
(x-5)^2 = 4*3.5(y-3.5)
(x-5)^2 = 14(y-3.5)