Question

parabolas

Answer 1

Find the standard form of the equation of the parabola with a focus at (0, -10) and a directrix at y = 10.

@aajugdar

Answer 2

Okay, first things first, standard-form:
\[\Large (x-\color{red}h)^2 = 4\color{blue}p(y-\color{green}k)\]

There are three coloured 'unknowns' and we must find them :D
\(\large (\color{red}h, \color{green}k) \) is the vertex of the parabola, while the value \(\large \color{blue}p\) is the directed distance from the focus to the vertex.

Sound daunting? ^^

Answer 3

you have focus given (0,-10) and directrix at y = 10
so

Answer 4

oh he is explainin ^^

Answer 5

Oh, sorry... you got this, @aajugdar ?

Answer 6

no
continue
i will check later

Answer 7

Anyway, @lovelyharmonics
you catch me so far, what I said? ^_^

Answer 8

no just kidding its -10

Answer 9

No... in fact, the vertex would be smack in the middle of the focus and the directrix... now, which point would that be? (lol, aajugdar's illustration should give you a hint ^_^ )

Answer 10

then it has to be 0

Answer 11

k = 0, yes
What about h? (the x-value of the vertex)

Answer 12

but.... (0,-10) would suggest it equals 0

Answer 13

In simpler terms, as you can see from the illustration, the vertex is simply the origin (0,0), right? :)

Answer 14

oh makes sense yeah

Answer 15

So, both h and k are zero, the best case (since it looks so much simpler that way)

\[\Large x^2 = 4\color{blue}py\]
We still have to find p, though.

How far is the vertex from the focus?

Answer 16

.-. how do you tell that?

Answer 17

Distance formula? lol
Vertex is (0,0) and focus is (0,-10)

I don't even think the distance formula is necessary XD

Answer 18

oh -10

Answer 19

Strictly speaking, the distance is 10 (distance is never negative)

BUT, here's the tricky part~ if the focus is BELOW the vertex (which it is) take p to be negative, and if the focus is ABOVE the vertex, then take p to be positive.

In this case, the focus (0, -10)... is it above or below the vertex (0,0) ?

Answer 20

scratch that, stupid question, I already answered it prior XD
So,
p = -10

And we're done ^^

Answer 21

so its -40x?

Answer 22

?...

Answer 23

it's -40y
:/

Answer 24

no its not...
its one of these
y = -1/10x2

y = -1/40x2

y2 = -40x

y2 = -10x

Answer 25

Oh you poor thing :D
(kidding)
\[\Large x^2 = -40y\]
What happens when you divide both sides by -40?

Answer 26

you get b

Answer 27

Yup.

There you go ^^

Answer 28

thank you c: i have like 20 more of these .-. and there slowing killing me