To find the directrix of the given parabola, we need to identify the properties and standard form of the parabola equation.
The given equation is:
[tex]\[ x^2 = -36y \][/tex]
This parabola opens downward since the coefficient of [tex]\( y \)[/tex] is negative. The standard form of a parabola that opens up or down is:
[tex]\[ (x - h)^2 = 4p(y - k) \][/tex]
For the given equation, we can identify that:
[tex]\[ x^2 = -36y \][/tex]
Here, the vertex [tex]\((h, k)\)[/tex] is at the origin, [tex]\((0, 0)\)[/tex]. Thus:
[tex]\[ x^2 = 4p(y - 0) \][/tex]
Equating the given equation [tex]\(x^2 = -36y\)[/tex] to the standard form [tex]\(x^2 = 4p(y - 0)\)[/tex], we have:
[tex]\[ 4p = -36 \][/tex]
Solving for [tex]\(p\)[/tex]:
[tex]\[ p = \frac{-36}{4} \][/tex]
[tex]\[ p = -9 \][/tex]
In this form, [tex]\(p\)[/tex] is the directed distance from the vertex to the focus and to the directrix. Because our [tex]\(p\)[/tex] is [tex]\(-9\)[/tex], it means the focus [tex]\(p\)[/tex] units downward from the vertex (since the parabola opens downwards). The directrix is [tex]\(p\)[/tex] units upward from the vertex, which effectively cancels out the negative sign.
Hence, the equation for the directrix is given by:
[tex]\[ y = k - p \][/tex]
Since [tex]\(k = 0\)[/tex]:
[tex]\[ y = 0 - (-9) \][/tex]
[tex]\[ y = 9 \][/tex]
Therefore, the directrix of the parabola [tex]\(x^2 = -36y\)[/tex] is:
[tex]\[ y = 9 \][/tex]
