Can you help me with parabolas?

Question

I need help with this problem on parabolas...
Find the fous an directrex of the parabola whose equation is 2y^2 + 4y= 8 + x.

Thank you!

Answer 1

Find the focus and directrex of the parabola whose equation is:

2y² + 4y = 8 + x

Rewrite the equation in vertex form.

x + 8 = 2y² + 4y

x + 8 + 2 = 2y² + 4y + 2

x + 10 = 2(y² + 2y + 1)

x + 10 = 2(y + 1)²

(1/2)(x + 10) = (y + 1)²

4p = 1/2
p = 1/8

The vertex (h,k) = (-10, -1).

Since this is a horizontal parabola, the line of symmetry is a horizontal line thru the vertex. The line of symmetry is:

y = k = -1

The focus is inside the parabola on the line of symmetry at a directed distance of p from the vertex. The focus is:

(h + p, k) = (-10 + 1/8, -1) = (-79/8, -1)

The directrix is a line outside the parabola that is perpendicular to the line of symmetry at a directed distance of -p from the vertex. The directrix is:

x = h - p = -10 - 1/8 = -81/10

Answer 2

At the end of this Wikipedia article section are the equations for the focus and the directorix of any parabola.
http://en.wikipedia.org/wiki/Parabola#Derivation_of_the_focus
The variables are explained on the diagram.