Question

How do you find the focus, vertex, and directrix of `x^2 -6x -36y +225=0`?

Answer 1

The vertex is `(a,b)` `=(3,6)`
The focus is `(a,b+p/2)` `=(3,15)`
And the directrix is `y=b-p/2` ; `y=6-9=-3`

Explanation:

Let's rewrite the equation
`x^2-6x-36y+225=0`
`x^2-6x+9=36y-225+9`
`(x-3)^2=36y-216=36(y-6)`
This is the equation of a parabola
`(x-a)^2=2p(y-b)`
`a=3`
`b=6`
`p=18`
The vertex is `(a,b)` `=(3,6)`
The focus is `(a,b+p/2)` `=(3,15)`
And the directrix is `y=b-p/2` ; `y=6-9=-3`

graph{((y-(x-3)^2/36-6)(y+3))=0 [-41.1, 41.07, -20.53, 20.6]}