Question

Focus on the negative y -axis, 6 units away from the directrix.

Short answer

The equation of the parabola with vertex at the origin, focus on the positive y axis and the focus 2 units from its directrix is given by $x^2=-12y$.

Step-by-step solution

Step 1. Given information.

Given: We have a parabola that has its vertex at the origin.

Step 2. We have to find the parabola that has its focus on the positive y  axis, 6 units away from the directrix.

We use the properties of the parabola to find the equation of the parabola.

Since the focus of the parabola is on they axis, the parabola is of the type $y^2=4py$

It is also given that the focus is on the positive y axis. Therefore, the parabola opens to the downwards with p<0.

The vertex of the parabola is at the point(0,0).

The coordinates of the focus and the directrix on the y axis is given by(0,-p) and(0,p) . The distance between the focus and the directrix is therefore equal to $\left|2p\right|$ units. So,

$\begin{array}{l}\Rightarrow \left|2p\right|=6\\ \Rightarrow p=3\text{\hspace{0.17em}and}\\ \Rightarrow p=-3\end{array}$

We take the negative value ofp as it opens to the downwards
Therefore the equation of the parabola is,

$\begin{array}{l}\Rightarrow x^2=4(-3)y\\ \Rightarrow x^2=-12y\\ \Rightarrow x^2=4(-3)y\\ \Rightarrow x^2=-12y\end{array}$