The parabola has a focus at [tex]\((-3,0)\)[/tex] and directrix [tex]\(x=3\)[/tex]. What is the correct equation for the parabola?

A. [tex]\(x^2=-12 y\)[/tex]

B. [tex]\(x^2=3 y\)[/tex]

C. [tex]\(y^2=3 x\)[/tex]

D. [tex]\(y^2=-12 x\)[/tex]



Answer :

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]