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.
Code:
| 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: \(\displaystyle \:x^2\;=\;8y\)
