Question

The focus of a parabola is (-6, -3). The directrix of the parabola is y =4.

What is the equation of the parabola?
o y = - a (x + 6)? +
o y = - (x + 6)+
o y = - + (x + 1)² +6
y = -5(x - ) - 6
14
2
1
2

Answer 1

well, we know the directrix is at y = 4 and the focus point is down below it at (-6 , -3), meaning is a downward opening vertical parabola, like the one you see in the picture below.

keeping in mind that the vertex is equidistant from the directrix and focus point by a distance "P", that puts this vertex at ( -6 , 1/2 ), and since the parabola is opening downwards, P is negative.

`\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=-6\\ k=(1)/(2)\\[1em] P=-(7)/(2) \end{cases}\implies 4\left( -\cfrac{7}{2} \right)\left(y-\cfrac{1}{2} \right)=[x-(-6)]^2 \\\\\\ -14\left(y-\cfrac{1}{2} \right)=(x+6)^2\implies y-\cfrac{1}{2}=-\cfrac{1}{14}(x+6)^2 \\\\[-0.35em] ~\dotfill\\\\ ~\hfill y=-\cfrac{1}{14}(x+6)^2+\cfrac{1}{2}~\hfill`