Question

writing equations of parabolas
the parabola has a focus at (-3, 0) and directrix x = 3. what is the correct equation for the parabola?
○ $x^{2}=-12y$
○ $x^{2}=3y$
○ $y^{2}=3x$
○ $y^{2}=-12x$

Explanation:

Step1: Recall parabola formula

For a parabola with a horizontal axis of symmetry, the standard - form equation is $(y - k)^2=4p(x - h)$, where $(h,k)$ is the vertex and $p$ is the distance from the vertex to the focus (or from the vertex to the directrix). The vertex of a parabola is the mid - point between the focus and the directrix.

Step2: Find the vertex

The focus is at $(-3,0)$ and the directrix is $x = 3$. The $x$ - coordinate of the vertex $h=\frac{-3 + 3}{2}=0$, and the $y$ - coordinate of the vertex $k = 0$. So the vertex is at $(0,0)$.

Step3: Calculate the value of $p$

The distance $p$ from the vertex $(0,0)$ to the focus $(-3,0)$ is $p=-3$ (negative because the focus is to the left of the vertex for a parabola opening left).

Step4: Write the equation

Substitute $h = 0$, $k = 0$, and $p=-3$ into the equation $(y - k)^2=4p(x - h)$. We get $y^{2}=4\times(-3)x=-12x$.

Answer:

$y^{2}=-12x$