Question

How do you find an equation of the parabola with focus (2,2) and directrix x=-2?

Answer 1

`(y-2)^2=16(x-2)`

Explanation:

The key to solving this problem is knowing all your relevant equations. The formula for a parabola in vertex form is:

`(y-k)^2=4p(x-h)`
Vertex: `(h, k)`
Focus: `(p+h, k)`
Directrix: `x=-p+h`

Notice we chose the version that squares the `y`-value. This is because the given directrix is `x=-2`. If the directrix had been `y=-2`, then the formulas are different.

You are given the focus of `(2,2)`
`(h, k)=(2, 2)`
So `h=2` and `k=2`

You are also given the directrix `x=-2`
`x=-p+h=-2`

We already determined that `h=2` from above, so
`-p+2=-2`
`-p=-4` (subtract 2 from both sides)
`p=4` (divide both sides by -1)

Plugging these values back into the original equation gives
`(y-2)^2=4(4)(x-2)`
`(y-2)^2=16(x-2)`