Question

Help please!
1) A comet that passes through the solar system just once has a path that is modeled by a conic section. Astronomers discover a comet whose path is modeled by the following equation, with the Sun at one focus. What type of conic section is the comet's path?
x^2/2,500-y^2/37,500=1

2) What are the vertices of the comet's path? (Hint: Examine the general formula for the path and identify the numbers that determine the vertices.)

3) At its closest point to the Sun, this comet is ____ million miles from the Sun. (Enter only the number.)
Hint: Subtract the vertex from the focus

Thank you!

Answer 1

1. Hint: The negative sign gives it away.

Answer 2

Negative sign? @Mertsj

Answer 3

Yes. do you see the minus sign between the terms on the left side?

Answer 4

On number 1? Yes @Mertsj

Answer 5

It identifies the conic section for you if you look at the general equations.

Answer 6

Which conic section is it?

Answer 7

Um... ellipse?

Answer 8

http://math2.org/math/algebra/conics.htm

Answer 9

Scroll to the bottom of that page and look for the one that looks like your equation.

Answer 10

Hyperbola? @Mertsj

Answer 11

Good for you!!

Answer 12

Wooo hooo! :) @Mertsj
Can you help with the other two please?

Answer 13

Before you find the vertices, you need to identify the center. Can you do that?

Answer 14

Um... I don't think so. I have a hard time with math.

Answer 15

\[\frac{(x-h)^2}{a^2}-\frac{(x-k)^2}{b^2}=1\]

Answer 16

In that equation (h,k) is the center. Can you match your equation up with that one and identify h and k?

Answer 17

Would they just both be 1?

Answer 18

I think the reason they aren't there is because they are both 0. So the center is (0,0)

Answer 19

Oh okay

Answer 20

And now you need to identify a^2

Answer 21

a^2 is 2,500

Answer 22

a^2 is the number under x^2. what is it?

Answer 23

And so yes it is 2500 so what is a?

Answer 24

50

Answer 25

Yes. Now a is important because the vertices are a units to the right and a units to the left of the center. So what are the vertices?

Answer 26

(50, 0) and (-50, 0)

Answer 27

Good.

Answer 28

Okay. :)

Answer 29

Now we need to identify b.

Answer 30

What is it?

Answer 31

Um... i got 193.64

Answer 32

me too.

Answer 33

Now find c by using the equation: a^2+b^2=c^2

Answer 34

We need c so we can find the foci.

Answer 35

I got 40000

Answer 36

That would be c^2. Right?

Answer 37

Yeah

Answer 38

So c is?

Answer 39

200

Answer 40

c is important because the foci are c units to the right and c units to the left of the center.

Answer 41

And now you can apply the hint given in 3.

Answer 42

Wait... hold on... I'm confused

Answer 43

The hint says to subtract the vertex from the focus. What is 200-50?

Answer 44

Oh. 150

Answer 45

So the closest the comet comes to the sun is 150 million miles.

Answer 46

Okay. Cool! If I could hug you right now I would! You are so awesome! Thanks!!!!

Answer 47

You are welcome.