issue interpreting the question... (parabola)

[shawie]

I am given the focus and the directrix and this problem is asking me to write and simplify an equation that specifies the set of points P(x,y) that are equidistant from F and d.

i'm guessing they want me to write an equation with the focus at (0,2) and a directrix y=-2 ...but what do they mean and simplify... i'm confused

 

[jsbeckton]

that means plug the information given into the parabola formula and then simplify.

 

[soroban]

Hello, shawie!

I am given the focus and the directrix and this problem is asking me to:
write and simplify an equation that specifies the set of points P(x,y) that are equidistant from F and d.

i'm guessing they want me to write an equation with the focus at (0,2) and a directrix y=-2
...but what do they mean and simplify>

They want you to derive the equation.

          |           P
          |           *(x,y)
          |       *   :
          |   *       :
     (0,2)*F          :
          |           :
    ------+-----------:--
          |           :
    - - - + - - - - - +  d
        -2|

The distance from point $\displaystyle P(x,y)$ to the focus $\displaystyle F(0,2)$ is:

$\displaystyle \;\;\;PF\;=\;\sqrt{(x\,-\,0)^2\,+\,(y\,-\,2)^2}$


The distance from point $\displaystyle P(x,y)$ to the directrix $\displaystyle y\,=\,-2$ is:

$\displaystyle \;\;\;Pd\;=\;y\,+\,2$


The two distances are equal: $\displaystyle \;\sqrt{x^2\,+\,(y\,-\,2)^2}\;=\;y\,+\,2$

$\displaystyle \;\;$That is the equation . . . but they want it simplified . . .


Square both sides: $\displaystyle \;x^2\,+\,(y\,-\,2)^2\;=\;(y\,+\,2)^2$

Expand: $\displaystyle \;x^2\,+\,y^2\,-4y\,+\,4\;=\;y^2\,+\,4y\,+\,4$

$\displaystyle \;\;$which simplifes to: $\displaystyle \:x^2\;=\;8y$