Find Parabola Given Focus & Directrix - Help Needed

In summary, the formula for the parabola with a focus at (-5, -5) and a directrix of y = 7 is (x+5)^2 = -24(y-1).
  • #1
MarkFL
Gold Member
MHB
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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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  • #2
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:

\(\displaystyle y=-\frac{x^2+10x+1}{24}\)
 
  • #3
Hello, Abs!

Find the equation of the parabola with focus at (-5, -5)
and directrix [tex]y = 7. [/tex]
Code:
                    |
                    |7
          - - . - - + - - -
              :     |
              :V    |
              o     |
    - - - * - : - * + - - - - -
        *     :     *
       *      o     |*
              :F    |
      *       :     | *
                    |
The focus [tex](F)[/tex] is (-5,-5).
The vertex [tex](V)[/tex] is (-5,1).

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

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

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

FAQ: Find Parabola Given Focus & Directrix - Help Needed

1. What is a parabola?

A parabola is a U-shaped curve that is created by the intersection of a plane and a cone. It is a type of conic section, along with circles, ellipses, and hyperbolas.

2. What is a focus and directrix?

A focus is a fixed point inside the parabola, while the directrix is a fixed line outside the parabola. These two elements determine the shape and position of the parabola.

3. How do I find the equation of a parabola given the focus and directrix?

To find the equation, you can use the distance formula to calculate the distance from a point on the parabola to the focus and to the directrix. Set these two distances equal to each other and then simplify the equation to the standard form of a parabola, y = ax^2 + bx + c.

4. Can a parabola have more than one focus and directrix?

No, a parabola can only have one focus and one directrix. These two elements are fixed and unique for each parabola.

5. How can knowing the focus and directrix of a parabola be useful?

Knowing the focus and directrix can help determine the shape and position of the parabola, as well as its equation. This information can be useful in various fields such as physics, engineering, and mathematics.

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