Given:
Focus of a parabola is (0, -12) and directrix is y = 12.
To find:
The equation of the parabola.
Solution:
General form of a parabola is
[tex](x-h)^2=4p(y-k)[/tex] ...(i)
where, (h,k) is vertex, (h,k+p) is focus and y=k-p is directrix.
Focus is (0, -12).
[tex](h,k+p)=(0,-12)[/tex]
[tex]h=0[/tex]
[tex]k+p=-12[/tex] ...(ii)
Directrix is y = 12.
[tex]k-p=12[/tex] ...(iii)
Adding (ii) and (iii), we get
[tex]2k=0[/tex]
[tex]k=0[/tex]
Putting k=0 in (ii), we get
[tex]0+p=-12[/tex]
[tex]p=-12[/tex]
Putting h=0, k=0 and p=-12 in (i).
[tex](x-0)^2=4(-12)(y-0)[/tex]
[tex]x^2=-48y[/tex]
[tex]-\dfrac{1}{48}x^2=y[/tex]
Therefore, the required equation of parabola is [tex]y=-\dfrac{1}{48}x^2[/tex].
