Find the equation of a parabola with focus (2, -3) and directrix x = 5.

Yay

directix x=5
that means the parabola opens left or right

the equation is
(y-h)²=4p(x-h)

alrighty
vertex is (h,k)
p is distance from focus to vertex
also distance from vertex to directix

if p>0 then the parabola opens to the right and focus is to the right of directix
if p<0 then the parabola opens to the left and focus is to the left of directix

alrighty
the distance between directix and focus divided by 2=p
hmm, note which one is to the left or right of which

directix is x=5
focus is (2,-3) where x=2
focus is to the left of directix
alrighty
fro m 5 to 2 the distance is 3
3/2=1.5
and since the focus is to the left of the directix, it is -1.5

so the parabols opens to the left
the vertex is 1.5 untis to the right of the focus
so vertex is at (3.5,-3)

so
equation is
(y-(-3))²=4(-1.5)(x-3.5)
(y+3)²=4(-1.5)(x-3.5)