Question

What is the vertex form of the equation of the parabola with a focus at (8,7) and a directrix of `y=18`?

Answer 1

`y=-1/22(x-8)^2+25/2`

Explanation:

Let their be a point `(x,y)` on parabola. Its distance from focus at `(8,7)` is

`sqrt((x-8)^2+(y-7)^2)`

and its distance from directrix `y=18` will be `|y-18|`

Hence equation would be

`sqrt((x-8)^2+(y-7)^2)=(y-18)` or

`(x-8)^2+(y-7)^2=(y-18)^2` or

`x^2-16x+64+y^2-14y+49=y^2-36y+324` or

`x^2-16x+22y-211=0`

or `22y=-x^2+16x+211`

or `y=-1/22(x^2-16x+64)+211/22+64/22`

or `y=-1/22(x-8)^2+275/22`

or `y=-1/22(x-8)^2+25/2`

graph{y=-1/22(x-8)^2+25/2 [-31.84, 48.16, -12.16, 27.84]}