Question

An archway will be constructed over a walkway. A piece of wood will need to be curved to match a parabola. Explain to Maurice how to find the equation of the parabola given the focal point and the directrix.
The directrix is at 9 and the focus is at (7, 6)

Answer 1

Ok

so the focus is on the opening side of the parabola

so for a vertex of (h,k)
opening up and down
(x-h)²=4p(y-k)
p is the distance from vertex to focus
it is also the distance from the vertex to the closes point on directix
2p=distance from focus to directix

p is positive if the focus is above the directix

ok
y=9 and (7,6)
9>6
directix is above the vertex
p is negative
find distance
from 9 to 6 is 3
3=2p
1.5=p
p is negative
and the vertex will be 1.5 above the focus and 1.5 below the directix
vertex is (7,7.5)

(x-7)²=4(-1.5)(y-7.5) is de equation