# Abs' question at Yahoo! Answers regarding finding a parabola given the focus and directrix

#### MarkFL

Staff member
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

Staff member
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

##### Well-known member
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: .$$(x+5)^2 \:=\:-24(y-1)$$