MHB Find Parabola Given Focus & Directrix - Help Needed

[MarkFL]

Here is the question:

Find the formula of this parabola?


Derive the equation of the parabola with a focus at (-5, -5) and a directrix of y = 7.

So... I've tried this one over and over but can't seem to get the right answer. Help anyone?

I have posted a link there to this topic so the OP can see my work.

 

[MarkFL]

Hello Abs,

A parabola is defined as the locus of all points $(x,y)$ equidistant from a point (the focus) and a line (the directrix). Using the square of the distance formula, we may write:

$$(x+5)^2+(y+5)^2=(y-7)^2$$

$$x^2+10x+25+y^2+10y+25=y^2-14y+49$$

Combining like terms, we obtain:

$$x^2+10x+1+24y=0$$

Solving for $y$, we get the quadratic function:

$$y=-\frac{x^2+10x+1}{24}$$

 

[soroban]

Hello, Abs!

Find the equation of the parabola with focus at (-5, -5)
and directrix y = 7.

                    |
                    |7
          - - . - - + - - -
              :     |
              :V    |
              o     |
    - - - * - : - * + - - - - -
        *     :     *
       *      o     |*
              :F    |
      *       :     | *
                    |

The focus (F) is (-5,-5).
The vertex (V) is (-5,1).

The form of this parabola is: (x-h)^2 \:=\:4p(y-k)
where (h,k) is the vertex,
and p is the directed distance from V to F.

We have: (h,k) = (-5,1) and p = -6.

The equation is: .(x+5)^2 \:=\:-24(y-1)