Question

Find an equation of the parabola with focus (-5, -2) and directrix y = 2.

Answer 1

Check the picture below, so the parabola looks more or less like so.

keeping in mind the vertex is half-way between the focus point and the directrix, this one lands as you see it in the picture, with a negative "p" distance.

`\textit{vertical parabola vertex form with focus point distance} \\\\ 4p(y- k)=(x- h)^2 \qquad \begin{cases} \stackrel{vertex}{(h,k)}\qquad \stackrel{focus~point}{(h,k+p)}\qquad \stackrel{directrix}{y=k-p}\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix}\\\\ \stackrel{`

`\begin{cases} h=-5\\ k=0\\ p=-2 \end{cases}\implies 4(-2)(y-0)=( ~~ x-(-5) ~~ )^2\implies -8y=(x+5)^2 \\\\\\ ~\hfill {\Large \begin{array}{llll} y=-\cfrac{1}{8}(x+5)^2 \end{array}} ~\hfill`