Question

find the vertex focus and directrix of the parabola x = 1/4 (y^2 + 2y + 33)

Answer 1

The standard form of the parabola equation is
\[(y-k)^{2}=4p(x-h)\]
so transforming x = 1/4 (y^2 + 2y + 33) to:
x=1/4[(y^2 + 2y + 1) + 32)]
x=14[ (y+1)^2 + 32]
\[(y+1)^2 = 4*1*(x-8)\]
so k = -1. h = 8 , p =1
1. vertex is (h,k) = (8,-1)
2. focus is (h,k+p) = (8,-1+1) = (8,0)
3. directrix is the equation y=k-p, so y=-2