How do you find the vertex, the focus, and the directrix of the parabola x^2 + 10x +8y +17 = 0?

1 Answer
Jun 24, 2018

Our first step is getting the equation into a form that easier to interpret.

Explanation:

We can do this by completing the square:

x^2+10x+8y=-17
(x^2+10x+25)+8y=-17+25
(x+5)^2+8y=8
8y=-(x+5)^2+8
y=-1/8(x+5)^2+1

The vertex is (-5,1) because the vertex, in this form of the equation, is (-h, k). -5 is -h and 1 is k.

To find the focus and the directrix, we need to know p, which is the distance from the vertex to both the focus and the directrix. We can find p by setting the scale factor (-1/8) equal to -1/(4p):

-1/8=-1/(4p)
8=4p
2=p

Before we find the focus and the directrix, we must remember that the graph faces down and that the vertex is (-5,1). The focus will be 2 units down from the vertex since the focus is always within the area covered by the parabola. The focus, therefore, is (-5,-1). The directrix is the line 2 units away from the vertex in the opposite direction. Since the graph faces down, the line will be horizontal and at y=3. Here's the graph of our equation in case you're confused:

graph{-1/8(x+5)^2+1 [-15, 5, -6, 4]}