To find the equation of the parabola with a given focus and directrix, we need to follow several steps:
1. Identify the focus and directrix:
- The given focus is at [tex]\((-3, 0)\)[/tex].
- The given directrix is the vertical line [tex]\(x = 3\)[/tex].
2. Determine the vertex of the parabola:
- The vertex lies exactly midway between the focus and the directrix.
- The x-coordinate of the vertex is the midpoint between the x-coordinate of the focus [tex]\((-3)\)[/tex] and the directrix [tex]\(3\)[/tex].
[tex]\[
\text{Vertex } (h, k) = \left( \frac{-3 + 3}{2}, 0 \right) = (0, 0)
\][/tex]
3. Determine the distance [tex]\(p\)[/tex] from the vertex to the focus:
- This distance is the absolute difference between the x-coordinate of the vertex and the x-coordinate of the focus.
[tex]\[
p = |0 - (-3)| = 3
\][/tex]
4. Formulate the equation of the parabola:
- The standard form of a parabola with the vertex at [tex]\((h, k)\)[/tex] and distance [tex]\(p\)[/tex] from the vertex to the focus is:
[tex]\((y - k)^2 = 4p(x - h)\)[/tex] for horizontal parabolas.
- Since the y-coordinate [tex]\(k = 0\)[/tex] and x-coordinate [tex]\(h = 0\)[/tex], the equation simplifies to:
[tex]\[
y^2 = 4p \cdot x
\][/tex]
5. Substitute [tex]\(p = -3\)[/tex] to match the direction:
- Because the focus [tex]\((-3, 0)\)[/tex] is to the left of the vertex [tex]\((0, 0)\)[/tex], the parabola opens to the left. Hence [tex]\(p\)[/tex] is negative.
[tex]\[
y^2 = 4(-3)x = -12x
\][/tex]
So, the correct equation of the parabola is:
[tex]\[
y^2 = -12x
\][/tex]
Thus, the correct option is:
[tex]\[
y^2 = -12x
\][/tex]
