For the parabola with the given equation [tex]y=\frac{1}{12} x^2[/tex]
The vertex of the parabola = (0, 0)
The focus of the parabola = (0, 3)
The directrix of the parabola: y = -3
The given equation of the parabola is:
[tex]y=\frac{1}{12} x^2[/tex]
The vertex form of the equation of a parabola is given as:
[tex]y=a(x-h)^2+k[/tex]
Comparing the vertex form of the equation of a parabola to the equation [tex]y=\frac{1}{12} x^2[/tex]
The vertex, (h, k) = (0, 0)
The focus of the parabola = [tex](h, \frac{1}{4a} )[/tex]
h = 0, [tex]\frac{1}{4a} = \frac{1}{4(\frac{1}{12}) } = 3[/tex]
The focus of the parabola = (0, 3)
To find the directrix of the parabola, rewrite the equation of the parabola as:
[tex]x^2=12y[/tex]
Compare [tex]x^2=12y[/tex] with [tex]x^2=4ay[/tex]:
4a = 12
a = 12/4
a = 3
The directrix is given as:
y = -a
y = -3
The directrix: y = -3
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