Question

What is the equation of the parabola with a focus at (-5,-8) and a directrix of y= -3?

Answer 1

`y=-1/10x^2-x-8`

Explanation:

Parabola is the path traced by a point so that it's distance from a given point called focus and a given line called directrix is always equal.

Let the point on parabola be `(x,y)`.

It's distance from focus `(-5,-8)` is `sqrt((x+5)^2+(y+8)^2)` and it's distance from line `y=-3` or `y+3=0` is `|y+3|`.

Hence the equation of the parabola with a focus at `(-5,-8)` and a directrix of `y= -3?` is

`sqrt((x+5)^2+(y+8)^2)=|y+3|`

or `(x+5)^2+(y+8)^2)=(y+3)^2`

or `x^2+10x+25+y^2+16y+64=y^2+6y+9`

or `10y=-x^2-10x-80`

or `y=-1/10x^2-x-8`

graph{ (10y+x^2+10x+80)(y+3)((x+5)^2+(y+8)^2-0.1)=0 [-15, 5, -10, 0]}