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:

\(\displaystyle (x+5)^2+(y+5)^2=(y-7)^2\)

\(\displaystyle x^2+10x+25+y^2+10y+25=y^2-14y+49\)

Combining like terms, we obtain:

\(\displaystyle x^2+10x+1+24y=0\)

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

\(\displaystyle y=-\frac{x^2+10x+1}{24}\)

 

[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: .$$(x+5)^2 \:=\:-24(y-1)$$