Question

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

Answer 1

(-5,5)

Answer 2

f(x) = −one twelfth (x − 5)2 + 2

f(x) = one twelfth (x − 5)2 + 2

f(x) = −one twelfth (x + 5)2 + 2

f(x) = one twelfth (x + 5)2 + 2

Answer 3

all those diamond things are supposed to be ( - ), i don't know why they show up like that

Answer 4

thought it was the third one but it was wrong too

Answer 5

Given two points, the focus (-5,5) and the directrix (x, -1), plug both points into this formula:

\((x - x_1)^2 + (y - y_1)^2 = (x - x_2)^2 + (y - y_2)^2\)

Then simplify

Answer 6

\[(-5 - x)^2 + (5 - 1)^2 = (-5 - x)^2 + (5 - 1)^2 \]

What did I do wrong here? I come up with zero

Answer 7

You're only supposed to plug the values for \((x_1, y_1)\) and \((x_2, y_2)\)

Answer 8

\((x_1, y_1)\) = (-5, 5)
\((x_2, y_2)\) = (x, -1)

Answer 9

Ok so
(x - x1)^2 + (y - y1)^2 = (x - x2)^2 + (y - y2)^2
(x - 5)^2 + (y - 5)^2 = (x - x)^2 + (y - 1)^2

Is it like this
\[(x - 5)^2 + (y - 5)^2 = (x - x)^2 + (y - 1)^2 \]
or this
\[(x + 5)^2 + (y - 5)^2 = (x - x)^2 + (y + 1)^2\]

I feel like I'm plugging these in wrong. :/

Answer 10

\((x + 5)^2 + (y - 5)^2 = (x - x)^2 + (y + 1)^2\) is correct. Try to understand why.

Answer 11

Because two ( - ) make a ( + )

Answer 12

Okay, now what can we do next?

Answer 13

simplify

Answer 14

Simplify what?

Answer 15

Umm the expression? or it doesn't need simplifying?

Answer 16

It has to be simplified. I was just asking about your simplification plan? How are you going to approach simplifying this?

Answer 17

@CrazyTurtle483, you still here?

Answer 18

Yes sorry my dad wanted me to go get something

Answer 19

Ok honestly i really tried to simplify this but i dont understand where to start
\[(x + 5)^2 + (y - 5)^2 = (x - x)^2 + (y + 1)^2 \]

Answer 20

+ x or do a +5?

Answer 21

Start with \((x - x)^2\)

Obviously x - x = 0

Answer 22

makes sense

Answer 23

and 0^2 = 0

Answer 24

Next, subtract \((y - 5)^2\) from both sides

Answer 25

(x + 5)^2 + (y - 5)^2 = (y + 1)^2
- (y - 5)^2 - (y - 5)^2
(x + 5)^2 = 12y − 24?

Answer 26

\[(x + 5)^2 = 12y - 24\]
I don't know why there are diamonds

Answer 27

Show me how you got from \((y + 1)^2 - (y - 5)^2\) to 12y - 24

Answer 28

I didn't know what (y+1)^2 - (y-5)^2 was so i distributed and combined like terms

Answer 29

What do you mean? Explain your algebraic steps.

Answer 30

since youre asking about 12y -24, is that wrong?
i dont know what (y+1)^2 - (y-5)^2 equals..

distribute
y^2 + 2y + 1 + − y^2 + 10y + −25

then you combine the like terms

(2y + 10y) + (y2 - y2) + (1 + -25)
12y + -24

Answer 31

Actually, instead of expanding what you do is use the difference of squares formula

Answer 32

\[y^2 + 2y + 1 + −y^2 + 10y + −25 \]

Answer 33

i havent learned that yet

Answer 34

Are you saying you don't know how to factor \(y^2 - 4\) ?

Answer 35

(y + 2) (y - 2) ?

Answer 36

Exactly I know that you know the difference of squares

Answer 37

didn't know factoring was also called difference of squares haha

Answer 38

ok so \[(x+5)^2 = (y+2)(y-2) ? \]

Answer 39

Difference of Squares is

\(a^2 - b^2 = (a + b)(a - b)\)

Answer 40

\(y^2 - 4 = y^2 - 2^2 = (y + 2)(y - 2)\)

Answer 41

\((y + 1)^2 - (y - 5)^2 = (y + 1 + y - 5)(y + 1 - (y - 5))\)

Answer 42

and \((y + 1 + y - 5))(y + 1 - (y - 5)) = (2y - 4)(6) = 12y - 24\)

Answer 43

Either way, 12y - 24 is correct

Answer 44

Now all you have to do is just continue solving for y

Answer 45

ok so would it be 6(2y - 4) = 12y - 24

Answer 46

then what because don't you distribute next?

Answer 47

Isolate y

Answer 48

What you have left is

\((x + 5)^2 = 12y - 24\)

Answer 49

ok but then what, isn't y isolated? or 12y - 24
y = 2

Answer 50

You don't know how to isolate y from there?

Answer 51

Add 24 to both sides. Afterwards, divide both sides by 12

Answer 52

(x+5)^2 = 12y - 24
+24 +24
ok so youre left with 12y one one side but what is (x+5)^2 + 24?

Answer 53

i feel like i should know this better

Answer 54

\[\frac{ 1 }{ 12 } (x+5)^{2} + 2 ? \]

Answer 55

you there @Hero

Answer 56

Looks right

Answer 57

Yay it was!

Answer 58

thank you so much man, you are one of the most patient people ive ever met

Answer 59

We don't stop until it's done and right.

Answer 60

Most people would probably have given up on me haha, thanks again.

Answer 61

No, it wasn't that bad.