What is the equation of the parabola with a focus at (-3, 0) and a directrix x = 3. a) y equals one-twelfth times x squared b) x equals negative one-twelfth times y squared c) y equals negative one-twelfth times x squared d) x equals one-twelfth times y squared

Given that the directrix of the parabola is x = 3 which is a vertical line, this means that the parabola opens up sideways (to the left or to the right).

Since the focus of the parabola is at (-3, 0) which is to the left of the directrix, this means that the parabola opens up to the left.

The vertex of the parabola is midway between the directrix and the focus.

Thus, the vertex of the parabola is (0, 0)

The equation of a parabola which opens to the left with the vertex at (h, k) is given by

[tex]-4p(x-h)=(y-k)^2[/tex]

where p is the distance between the vertex and the focus.

Given that the vertex is at (0, 0) and that the focus is at (-3, 0), the distance between point (0, 0) and (-3, 0) is 3 and thus p = 3.

Thus, the equation of the required equation is given by

[tex]-4(3)(x-0)=(y-0)^2 \\ \\ \Rightarrow-12x=y^2 \\ \\ \Rightarrow x=- \frac{1}{12} y^2[/tex]

Therefore, option b is the correct answer.