Refer to the attachment
koi h

A parabola in analytical geometry is a curve drawn that has either 2 x-intercepts or y-intercepts. You can see visually that it is a parabola if it forms an arc that passes either the x-axis or y-axis twice. Nevertheless, it would have two roots, so the equation must be quadratic.There are 2 possible equations for a parabola:

(x-h)² = +/- 4a(y-k) or
(y-k)² = +/- 4a(x-h), where

(h,k) are the coordinated of the vertex
a is the distance of the focus from the vertex

From the problem, it says that the vertex is at the origin, so the coordinates are (0,0). That means h=0 and k=0. Also, the focus is a=3. Thus, the possible equations for the reflector are

x² = +/- 12y and y² = +/- 12x

Any of those are possible because the problem mentions that it can open anywhere. Therefore, it could open downwards(-) or upwards(+) if it passes the x-axis twice; or to the left(-) or to the right(+) if its passes the y-axis twice.