Question

If I'm given:
Vertices at (-2,5) and (-2,1)
Foci (-2,6) and (-2,0)
then how do I find the equation of the hyperbole?

Answer 1

If you know the vertices and foci you can determine the orientation of the hyperbola. So do that first. Does it open right and left or up and down?

Answer 2

up and down

Answer 3

So the center is on the line x = -2 half way between the vertices. What is the center?

Answer 4

(-2,3) right?

Answer 5

Yes. And since it opens up and down, we know that y is the positive term. Do you agree?

Answer 6

yes. absolutely

Answer 7

Hang on a sec. Just had an interruption.

Answer 8

of course :)

Answer 9

Sorry.

Answer 10

So now we can write the numerators of our two fractions because we know the center. Can you do that?

Answer 11

one sec, this is like a refresher for me and a new subject for the person I'm with, so I'm going to talk her through this next part, just a second

Answer 12

ok

Answer 13

The numerators are \[3y^2 and -2x^2\] correct?

Answer 14

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

Answer 15

(h,k) is the center

Answer 16

oh okay, that's the best standard equation I've found all day. I love this site.

Answer 17

So now, if we can just find "a" and "b" we'll be done.

Answer 18

And remember a is the distance from the center to the vertex.

Answer 19

b is vertex to focus correct?

Answer 20

b is the distance from the center to the co-vertex (which we don't know) and c is the distance from the center to the focal point.
Fortunately we know that a, b, and c are related by this equation: a^2+b^2=c^2

Answer 21

So what is a? What is c?

Answer 22

9-4 so b is sqrt(5)

Answer 23

uh a^2 is 9? thought it was 4

Answer 24

oh you meant c

Answer 25

2^2+b^2=3^2

Answer 26

\[\frac{ (y-3)^2 }{ 4 } -\frac{ (x+2) }{ 5 } = 3\] is what I got

Answer 27

it's all good except the right side should be =1

Answer 28

okay, thank you :) big help, it's always 1?

Answer 29

Yes. When the equation is in the "h,k" form

Answer 30

okay thanks

Answer 31

yw

Answer 32

if it opens left right, A still goes under the first term right?

Answer 33

Yes. a^2 under positive term.

Answer 34

ty

Answer 35

yw