MHB Find Parabola Given Focus & Directrix - Help Needed

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To derive the equation of a parabola with a focus at (-5, -5) and a directrix of y = 7, the relationship between the focus and directrix is used. The equation is established using the distance formula, leading to the quadratic function y = -\frac{x^2 + 10x + 1}{24}. The vertex of the parabola is located at (-5, 1), with the directed distance from the vertex to the focus being -6. Consequently, the final equation of the parabola is (x + 5)^2 = -24(y - 1). This provides a complete representation of the parabola based on the given parameters.
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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.
 
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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:

$$y=-\frac{x^2+10x+1}{24}$$
 
Hello, Abs!

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