Answer
(a)
Vertex: $V(2,3)$
Focus: $F(5,3)$
Directrix: $x=-1$
(b)
Work Step by Step
Equation of a parabola with horizontal axis and vertex at $(h,k)$:
$(y-k)^2=4p(x-h)$
$y^2-6y-12x+33=0$
$y^2-6y+9=12x-33+9=12x-24$
$(y-3)^2=12(x-2)$
$h=2$
$k=3$
Vertex: $V(2,3)$
$4p=12$
$p=3$. Parabola opens to the right.
The given equation can be obtained by shifting
$y^2=12x$
right 2 units and upwnward 3 units:
Focus: $F(p,0)=F(3,0)$
Directrix: $x=-p=-3$
Now, shift the vertex and the directrix $2$ unit to the right and 3 units upwnward:
Focus: $F(5,3)$
Directrix: $x=-1$
