Question

A parabola can be drawn given a focus of (9, -2)(9,−2) and a directrix of x=3x=3. Write the equation of the parabola in any form.

Answer 1

I will guess you meant:
A parabola can be drawn given a focus of (9, -2) and a directrix of x=3. Write the equation of the parabola in any form.

Using the actual definition:
let P(x,y) be any point on the parabola
Distance of P from focal point = distance of P to the directrix
√((x-9)^2 + (y+2)^2) = x - 3

square both sides and simplify
x^2 - 18x + 81 + y^2 + 4y + 4 = x^2 - 6x + 9
y^2 + 4y + 76 = 12x

Answer 2

Recall that the parabola
y^2 = 4px has
focus at (p,0)
directrix x = -p
The vertex is midway between the directrix and the focus.
So your parabola has a vertex at (6,-2) and p=3
That makes its equation
(y+2)^2 = 12(x-6)

confirmation is at
https://www.wolframalpha.com/input/?i=parabola+%28y%2B2%29%5E2+%3D+12%28x-6%29

Answer 3

A parabola can be drawn given a focus of (2,6) and a directrix of x=8. What can be said about the parabola?