Find the vertex focus and directrix of the parabola y2 6y

Find the vertex, focus, and directrix of the parabola. y^2 + 6y + 8x + 25 = 0

Solution

We know the generl equation of parabola is
4p(x - h) = (y - k)²
Here vertex= (h,k), p= distance from the vertex to the focus = The distance from the vertex to the directrix
Given y² + 6y + 8x + 25 = 0
=> 8x + 25 = -y² - 6y
=> 8x + 25 - 9 = -(y² + 6y + 9)
=> 8x + 16 = -(y + 3)²
=> -8(x + 2) = (y + 3)²
So, the vertex (h,k) is (-2,-3).
The line of symmetry is a horizontal line thru the vertex.
y = -3
and
4p = -8
=> p = -2
Now since the focus is a distance | p | = 2 from the vertex along the line of symmetry. It is inside the parabola.

Hence, the focus is
(h + p, k) = (-2 - 2, -3) = (-4, -3)
The directrix is also a distance | p | = 2 from the vertex but in the opposite direction. It is a line perpendicular to the line of symmetry and is outside the parabola. The directrix is a vertical line with the equation:
x = h - p = -2 + 2 = 0
=> x = 0