Question

How do you find the vertex, the focus, and the directrix of the parabola `x^2 -8x -28y -124=0`?

Answer 1

`(4,-5),(4,2),y=-12`

Explanation:

`"the translated forms of the equation of a parabola are"`

`•color(white)(x)(y-k)^2=4p(x-h)larrcolor(blue)" opens horizontally"`

`•color(white)(x)(x-h)^2=4p(y-k)larrcolor(blue)" opens vertically"`

`"in both cases the "color(magenta)"vertex "=(h,k)`

`"p is the distance from the vertex to the focus "`
`"and directrix"`

`color(blue)"completing the square"" on "x^2-8x`
`"and moving all other terms to the right"`

`x^2-8xcolor(red)(+16)=28y+124color(red)(+16)`

`(x-4)^2=28y+140`

`rArr(x-4)^2=28(y+5)`

`rArrcolor(magenta)"vertex "=(4,-5)`

`"the parabola opens vertically up "4p>0`

`4p=28rArrp=7`

`rArrcolor(red)"focus "=(4,-5+7)=(4,2)larr" above the vertex"`

`color(blue)"directrix is "y=(-5-7)=-12larr" below the vertex"`