To identify the focus and directrix of the equation [tex]\(x = -\frac{1}{18} y^2\)[/tex], we can follow these steps:
1. Standard Form Identification:
The given equation [tex]\(x = -\frac{1}{18} y^2\)[/tex] can be compared to the standard form of a parabola that opens left/right, which is given by [tex]\((y - k)^2 = 4p(x - h)\)[/tex].
2. Transform to the Standard Form:
Rewrite the given equation:
[tex]\[
x = -\frac{1}{18} y^2
\][/tex]
Observe that this can be rearranged as:
[tex]\[
y^2 = -18x
\][/tex]
3. Identify Parameters:
Comparing [tex]\((y - k)^2 = 4p(x - h)\)[/tex] with [tex]\(y^2 = -18x\)[/tex], we can see that:
[tex]\[
4p = -18
\][/tex]
Solving for [tex]\(p\)[/tex]:
[tex]\[
p = \frac{-18}{4} = -4.5
\][/tex]
4. Vertex Identification:
The vertex form [tex]\((y - k)^2 = 4p(x - h)\)[/tex] indicates that the vertex is [tex]\((h, k)\)[/tex]. Given our equation, the vertex is at [tex]\((0,0)\)[/tex], i.e., [tex]\(h = 0\)[/tex] and [tex]\(k = 0\)[/tex].
5. Focus Calculation:
For a parabola that opens to the left (negative [tex]\(p\)[/tex]), the focus is located at [tex]\((h + p, k)\)[/tex]. Substituting the values we have:
[tex]\[
\text{Focus} = (0 + (-4.5), 0) = (-4.5, 0)
\][/tex]
6. Directrix Calculation:
The directrix of a parabola that opens left/right is a vertical line and is given by the equation [tex]\(x = h - p\)[/tex]. Substituting the values:
[tex]\[
\text{Directrix} = 0 - (-4.5) = 4.5 \implies x = 4.5
\][/tex]
Combining these results, we have identified:
- The focus as [tex]\((-4.5, 0)\)[/tex]
- The directrix as [tex]\(x = 4.5\)[/tex]
Therefore, the correct answer to the question is:
[tex]\[
F\left( -\frac{9}{2}, 0 \right), \quad x = \frac{9}{2}
\][/tex]
