Derive the equation of the parabola with a focus at (−5, 5) and a directrix of y = -1.

Question

anonymous · Answer

|dw:1395585270860:dw|

anonymous · Answer

you need to know what it  looks like to start, then it is not too hard

anonymous · Answer

general form is $$4p(y-k)=(x-4)^2$$

anonymous · Answer

plug it in i guess

anonymous · Answer

first find $p$

anonymous · Answer

since from the picture it is clear it opens down, and you are given the vertex as $(-5,5)$ we are up to 
$$4p(y-5)=(x+5)^2$$

anonymous · Answer

$p$ is the distance from the vertex to the directrix, which is $4$

anonymous · Answer

but since it opens down, use 
$$-20(y-5)=(x+5)^2$$

anonymous · Answer

lets check it

anonymous · Answer

good thing i checked, my answer is wrong!

anonymous · Answer

one second let me find my mistake

anonymous · Answer

ok thx

anonymous · Answer

ooooh it is the focus that is $(-5,5)$ not the vertex!!

anonymous · Answer

ok lets start again
first of all my picture is junk
it should look like this 
|dw:1395586060441:dw|

anonymous · Answer

now it opens UP!!
the vertex is half way between the focus and the directrix 
that would be the point $(-5,2)$

anonymous · Answer

|dw:1395586175059:dw|

anonymous · Answer

now we get it is 
$$4p(y-2)=(x+5)^2$$ and we need $p$

anonymous · Answer

$p$ is still the distance between the focus and the directrix, which is $3$ so lets try 
$$12(y-2)^2=(x+5)^2$$

anonymous · Answer

anonymous · Answer

ok