[tex]y = \frac{1}{24} x^{2}[/tex]
Step-by-step explanation:
Since directrix is a horizontal line, this is a regular vertical parabola, where the x part is squared. The equation of a vertical parabola is:
(x - h)² = 4p(y - k)
where,
(h,k) are the coordinates of the vertex
p = distance from the vertex to the focus
So, we need to find out h, k and p and plug those values :
We know that the vertex of a parabola is halfway between focus and the directrix
. We know the focus (0,4) and directrix y = 6 therefore, vertex (h,k) = (0, 0)
Now, p = distance from the vertex to the focus
= distance from the (0,0) to the (0,4)
= 4
Plug all the values in equation:
[tex](x-0)^{2} = 4(4)(y-0)\\x^{2} = 24y[/tex]
Now, we rearrange to write this into standard form (y =[tex]ax^{2}+bx+c[/tex])
or,[tex]y = \frac{1}{24} x^{2}[/tex]
