Question

Can anyone help with this question?

Derive the equation of the parabola with a focus at (-7, 5) and a directrix of y = -11
a. 1/32(x+7)^2 - 3
b. -1/32(x + 7)^2 -3
c. -1/32(x-7)^2 - 3
d. 1/32(x-7)^2 -3

Answer 1

what would be the formula I would use?

Answer 2

@annamasreya, are you still here?

Answer 3

Yes Hero. Hi! I think this is a graphing functions question but don't know how to approach it. I am an online/homeschooled

Answer 4

In general if you are given a focus point \((\color\green{x_1}, \color\red{y_1})\) and a directrix point \((\color\green{x_2}, \color\red{y_2})\), you can insert both points in to the following formula:

\((x - \color\green{x_1})^2 + (y - \color\red{y_1})^2 = (x - \color\green{x_2})^2 + (y - \color\red{y_2})^2\)

Answer 5

Afterwards, you would simplify the equation accordingly. In this case, we are given:

Focus = (-7,5)
Directrix = (x, -11)

Answer 6

ok I am with you so far

Answer 7

what would we put in for the x on the second set of coordinates? something that would work in the formula?

Answer 8

And if we insert both points in to the equation we have:

\((x - \color\green{(-7)})^2 + (y - \color\red{(5)})^2 = (x - \color\green{(x)})^2 + (y - \color\red{(-11)})^2\)

Answer 9

ok and just factor them out to get a quadratic statement?

Answer 10

Basically, it simplifies to

\((x + 7)^2 + (x + 5)^2 = (0)^2 + (y + 11)^2\)

or just
\((x + 7)^2 + (y + 5)^2 = (y + 11)^2\)

Answer 11

At this point what you want to do is expand \((y + 5)^2\) and \((y + 11)^2\)

Answer 12

it's asking to get the f(x)? I would get y^2 +10y +25 and y^2 +22y + 121?

Answer 13

Okay, so what you have at this point is:

\((x + 7)^2 + y^2 + 10y + 25 = y^2 + 22y + 121\)

Answer 14

Now subtract \(y^2 + 10y + 25\) from both sides.

Answer 15

I think I would end up with (x+7)^2 = 12y +121?

Answer 16

or do I factor out the (x+7)^2?

Answer 17

You forgot to subtract 25 from 121

Answer 18

oops yep so 12y +96

Answer 19

So you now have

\((x + 7)^2 = 12y + 96\)

At this point what you want to do is isolate \(y\) as this was the ultimate goal.

Answer 20

Actually hang on a minute...

Answer 21

ok

Answer 22

We should have gotten:

\((x + 7)^2 + (y - 5)^2 = (y + 11)^2\)

Which expands to:
\((x + 7)^2 + y^2 - 10y + 25 = y^2 + 22y + 121\)

Answer 23

Now subtract \(y^2 - 10y + 25\) from both sides.

Answer 24

so (x+7)^2 = 12y + 96? I think?

Answer 25

Had you subtracted correctly, you would have gotten

\((x + 7)^2 = 32y + 96\)

Answer 26

From there, you can solve for y

Answer 27

so I missed adding the 10y's together. I see

Answer 28

that would have got my 32y?

Answer 29

Continue solving for y

Answer 30

so subtract 96 from both sides then we get (x+7)^2 -96 over 32 = y?

Answer 31

Yes, but it can be simplifed further

Answer 32

hmmm. (x+7)^2 -3 over 32????? : )!?

Answer 33

\(\dfrac{(x + 7)^2}{32} - 3 = y\)

Answer 34

That result should agree with one of your answer choices.

Answer 35

Yes! Thank you soooooo much! I feel like we were on a quest and we just slayed the dragon!!!!

Answer 36

Thank you and I hope you are a teacher because you can really help people!

Answer 37

Which answer choice is correct?

Answer 38

it's A.

Answer 39

Yes, A is correct.

Answer 40

Thank you for the help! Have a good one!