Question

What is the vertex form of the equation of the parabola with a focus at (15,-5) and a directrix of `y=7`?

Answer 1

Conic form: `(x-11)^2 = 16(y-5)`

Explanation:

We know this up/down parabola `=>` equation of directrix is `y=7`

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

Focus `(h+p, k)`

Directrix: `y= h-p`

Focus `(15, -5)` so
1) `h+p= 15` ; `color(red)(k= -5)`

Directrix: `y= 7`
2) `h-p=7`

Now we have system of equation (add both equations)
`h+p= 15`
`h--p = 7`
`=> 2h =22 =>` `color(red)(h= 11)`

Now we solve for `p`
`h+p= 15`
`(11)+p =15` `=> color(red)(p= 4)`

Conic form
`(x-color(red)11)^2 =4*color(red)(4)(y-color(red)5)`
Answer: `(x-11)^2 = 16(y-5)`