Question

Regan is trying to find the equation of a quadratic that has a focus of (−2, 5) and a directrix of y = 13. Describe to Regan your preferred method for deriving the equation. Make sure you use Regan's situation as a model to help her understand.

I am really, really confused so its going to be hell trying to get me to understand this fyi sorry

Answer 1

The definition of a parabola is that the distance of any point on it from the focus is equal to the distance of the point from the directrix.
So if (x,y) is a point on the parabola, then find the distance of (x,y) from the focus (-2,5).
Find the distance of (x,y) from y = 13. Equate them and you will have your quadratic equation.

Answer 2

\[
(x+2)^2 + (y-5)^2 = (y-13)^2
\]
Simplify above.

Answer 3

Should I FOIL all of them and combine like terms to simplify?

Answer 4

@ashaboo456, did you understand how he set that up to begin with?

Answer 5

a little bit

Answer 6

He inserted points (-2,5) and (x, 13) the distance formula:

\(\sqrt{(x - x_1)^2 + (y - y_2)^2} = \sqrt{(x - x_2)^2 + (y - y_2)^2}\)

Answer 7

Keep in mind that the directrix y = 13 can be represented as the point (x, 13) since no matter what value x is, y will always be 13.

Answer 8

So after inputting the points, you have:

\(\sqrt{(x - (-2))^2 + (y - 5)^2} = \sqrt{(x - x)^2 + (y - 13)^2}\)

Answer 9

You can then square both sides to get:

\((x - (-2))^2 + (y - 5)^2 = (x - x)^2 + (y - 13)^2\)

Answer 10

And then that simplifies to

\((x + 2)^2 + (y - 5)^2 = (y - 13)^2\)

Answer 11

Now, it is convenient at this point to re-write it as

\((x + 2)^2 = (y - 13)^2 - (y - 5)^2\)

Answer 12

From here, you can expand the RHS.

Answer 13

Expand the RHS?

Answer 14

RHS = Right Hand Side

Answer 15

So FOIL out both Y parts then, and combine like terms and all?

Answer 16

Yes, however, I'd rather say expand or multiply because I'm not a fan of the FOIL method.

Answer 17

After expanding the RHS, you should be able to solve for y.

Answer 18

so now I have this :

2y^2-36y+194=(x+2)^2

Is that right?

Answer 19

Well, I was worried about how you would end up simplifying

Answer 20

It is important to keep track of your signs after expansion. Otherwise you might add when you are supposed to subtract.

Answer 21

yeah im sorry its a big jumbled mess ugh

Answer 22

At some point you should have gotten:

\((x + 2)^2 = y^2 - 26y + 169 - (y^2 - 10y + 25)\)

Answer 23

Oh I did! But i combined the two, I must have messed up somewhere there?

Answer 24

After distributing that negative afront \(y^2 - 10y + 25\) you would then have

\((x + 2)^2 = y^2 - 26y + 169 - y^2 + 10y - 25\)

Answer 25

And then, a good thing to do after that is place like terms together like so:

\((x + 2)^2 = (y^2 - y^2) + (10y - 26y) + (169 - 25)\)

Answer 26

Doing that helps you keep track and avoid mistakes.

Answer 27

Oh, so then it would it be (x+2)^2=-16y+144 ?

Answer 28

and then I can square each side to get rid of the square on the X part thing?

Answer 29

Well, what you want to do is isolate y

Answer 30

Wouldn't squaring each side first help make it simpler when trying to isolate the y part?

Answer 31

Actually, it wouldn't.

Answer 32

Oh yeah never mind that was stupid the 16 isn't even positive ugh sorry

Answer 33

Even if it were, it still wouldn't have helped.

Answer 34

However, if you subtract 144 from both sides, then divide both sides by -16, then y will be isolated.

Answer 35

okay, so I will end up with (x+2)^2 over 16 +9 = y
?

Answer 36

Don't forget the negative
\(-\dfrac{(x + 2)^2}{16} + 9 = y\)

Answer 37

Oh okay. So is that the final answer, that equation I mean.

Answer 38

Thankfully, yes.

Answer 39

Thank you so much, this seriously helped me so much. You're fantastic !

Answer 40

You're most welcome :)