Question

What is the equation in standard form of the parabola with a focus at (-2,3) and a directrix of y= -9?

Answer 1

`y=(x^2)/24+x/6-17/6`

Explanation:

Sketch the directrix and focus (point `A` here) and sketch in the parabola.
Choose a general point on the parabola (called `B` here).
Join `AB` and drop a vertical line from `B` down to join the directrix at `C`.
A horizontal line from `A` to the line `BD` is also useful.

By the parabola definition, point `B` is equidistant from the point `A` and the directrix, so `AB` must equal `BC`.
Find expressions for the distances `AD`, `BD` and `BC` in terms of `x` or `y`.
`AD=x+2`
`BD=y-3`
`BC=y+9`
Then use Pythagoras to find AB:
`AB=sqrt((x+2)^2+(y-3)^2)`
and since `AB=BC` for this to be a parabola (and squaring for simplicity):
`(x+2)^2+(y-3)^2=(y+9)^2`
This is your parabola equation.
If you want it in explicit `y=...` form, expand the brackets and simplify to give `y=(x^2)/24+x/6-17/6`