Question

What is the vertex form of the equation of the parabola with a focus at (12,22) and a directrix of `y=11`?

Answer 1

`y=1/22(x-12)^2+33/2`

Explanation:

`"the equation of a parabola in "color(blue)"vertex form"` is.

`color(red)(bar(ul(|color(white)(2/2)color(black)(y=a(x-h)^2+k)color(white)(2/2)|)))`

`"where "(h,k)" are the coordinates of the vertex and a"`
`"is a multiplier"`

`"for any point "(x.y)" on a parabola"`

`"the focus and directrix are equidistant from "(x,y)`

`"using the "color(blue)"distance formula "" on "(x,y)" and "(12,22)`

`rArrsqrt((x-12)^2+(y-22)^2)=|y-11|`

`color(blue)"squaring both sides"`

`rArr(x-12)^2+(y-22)^2=(y-11)^2`

`(x-12)^2cancel(+y^2)-44y+484=cancel(y^2)-22y+121`

`rArr(x-12)^2 =22y-363`

`rArry=1/22(x-12)^2+33/2larrcolor(red)"in vertex form"`