Question

Given the directrix of y = 6 and focus of (0, 4), which is the equation of the parabola?

Answer 1

$y = \frac{1}{24} x^{2}$

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:

$(x-0)^{2} = 4(4)(y-0)\\x^{2} = 24y$

Now, we rearrange to write this into standard form (y =$ax^{2}+bx+c$)

or,$y = \frac{1}{24} x^{2}$