Question

What is the equation of the parabola with a focus at (-8,-4) and a directrix of y= 5?

Answer 1

`y=-1/18(x+8)^2-8/9`

Explanation:

Parabola is the locus of a point, which movesso that its distance from a point called focus and a line called directrix is always equal.

Let the point be `(x,y)`,

its distance from `(-8,-4)` is `sqrt((x+8)^2+(y+4)^2)`

and its distance from line `y=5` is `|y-5|`

Hence equation of parabola is `sqrt((x+8)^2+(y+4)^2)=|y-5|`

or `(y-5)^2=(x+8)^2+(y+4)^2`

or `y^2-10y+25=(x+8)^2+y^2+8y+16`

or `-10y-8y=(x+8)^2+16`

or `-18y=(x+8)^2+16`

or `y=-1/18(x+8)^2-8/9` (in vertex form)

graph{(y+1/18(x+8)^2-8/9)(y-5)((x+8)^2+(y+4)^2-0.09)=0 [-24.92, 15.08, -9.2, 10.8]}