We are given focus at (0, -5) and directrix y = 5.
The formula for equation of directrix is y = k - p.
And formula for focus is (h, k + p).
On comparing focus (0, -5) and (h, k + p), h=0.
k+p = -5 ----------------equation(1)
On comparing directrix y = k - p and y = 5.
k- p = 5 ----------------equation(2)
Adding eqautions (1) and (2).
k+p = -5
k- p = 5
_________
2k = 0.
k=0
Plugging k=0 in first equation, we get
k+p = -5
0 + p = -5.
p = -5.
We know, parabola equation (x - h)^2 = 4p (y - k)
Plugging h,k and p value in the above equation, we get
(x - 0)^2 = 4(-5) (y - 0)
x^2 = -20y.
Dividing both sides by -20, we get
-1/20 x^2 = y.
Or
[tex]y=-\frac{1}{20}x^2[/tex]
Therefore, the equation of the parabola is [tex]y=-\frac{1}{20}x^2[/tex].
