Answer:
The standard form of the equation of the parabola is [tex]y=-\frac{x^2}{-32}[/tex].
Step-by-step explanation:
The general form of a parabola is
[tex](x-h)^2=4p(y-k)[/tex]
Where, (h,k) is vertex, (h,k+p) is focus and y=k-p is directrix.
Focus of the parabola is (0, -8).
[tex](h,k+p)=(0,-8)[/tex]
[tex]h=0[/tex]
[tex]k+p=-8[/tex] .... (1)
Directrix of the parabola is
[tex]k-p=8[/tex] .... (2)
On adding (1) and (2) we get
[tex]2k=0[/tex]
[tex]k=0[/tex]
Put this value in equation (1).
[tex]0+p=-8[/tex]
[tex]p=-8[/tex]
The value of p is -8.
Substituent h=0,k=0 and p=-8 in general form of parabola.
[tex](x-0)^2=4(-8)(y-0)[/tex]
[tex]x^2=-32y[/tex]
Divide both sides by -32.
[tex]\frac{x^2}{-32}=y[/tex]
Therefore the standard form of the equation of the parabola is [tex]y=-\frac{x^2}{-32}[/tex].
