Question

Given the equation x^{2} = 8y what are the coordinates of the focus

Answer 1

Focus ->
`(0,2)`

Explanation:

Because the x-term is squared, the parabola must be vertical.

Recall that the equation for a vertical parabola is
`4p(y-k)=(x-h)^2` where
`p` is the distance from the vertex
`(h,k)` to the focus and
`(h,k+p)` are the coordinates of the focus.

We can tell by the given equation that the vertex must be located at the origin.

Therefore, we can determine the value of
`p` having known the vertex and the direction of the parabola:

`4p(y-k)=(x-h)^2`

`4p(y-0)=(x-0)^2`

`4py=x^2`

`4py=8y`

`4p=8`

`p=2`

So, given that
`p=2`, this means that the coordinates of the focus are
`(0,2)`.